Classification of transmission

ABSTRACT

Disclosed are various embodiments for distributing power to loads and classifying loads that receive electrical energy in the form of guided surface waves that are transmitted by guided surface waveguide probes along a terrestrial medium.

CROSS REFERENCE TO RELATED APPLICATIONS

This application is related to co-pending U.S. Non-provisional PatentApplication entitled “Excitation and Use of Guided Surface Wave Modes onLossy Media,” which was filed on Mar. 7, 2013 and assigned applicationSer. No. 13/789,538, and was published on Sep. 11, 2014 as PublicationNumber US2014/0252886 A1, and which is incorporated herein by referencein its entirety. This application is also related to co-pending U.S.Non-provisional Patent Application entitled “Excitation and Use ofGuided Surface Wave Modes on Lossy Media,” which was filed on Mar. 7,2013 and assigned application Ser. No. 13/789,525, and was published onSep. 11, 2014 as Publication Number US2014/0252865 A1, and which isincorporated herein by reference in its entirety. This application isfurther related to co-pending U.S. Non-provisional Patent Applicationentitled “Excitation and Use of Guided Surface Wave Modes on LossyMedia,” which was filed on Sep. 10, 2014 and assigned application Ser.No. 14/483,089, and which is incorporated herein by reference in itsentirety. This application is further related to co-pending U.S.Non-provisional Patent Application entitled “Excitation and Use ofGuided Surface Waves,” which was filed on Jun. 2, 2015 and assignedapplication Ser. No. 14/728,507, and which is incorporated herein byreference in its entirety. This application is further related toco-pending U.S. Non-provisional Patent Application entitled “Excitationand Use of Guided Surface Waves,” which was filed on Jun. 2, 2015 andassigned application Ser. No. 14/728,492, and which is incorporatedherein by reference in its entirety.

BACKGROUND

For over a century, signals transmitted by radio waves involvedradiation fields launched using conventional antenna structures. Incontrast to radio science, electrical power distribution systems in thelast century involved the transmission of energy guided along electricalconductors. This understanding of the distinction between radiofrequency (RF) and power transmission has existed since the early1900's.

BRIEF DESCRIPTION OF THE DRAWINGS

Many aspects of the present disclosure can be better understood withreference to the following drawings. The components in the drawings arenot necessarily to scale, emphasis instead being placed upon clearlyillustrating the principles of the disclosure. Moreover, in thedrawings, like reference numerals designate corresponding partsthroughout the several views.

FIG. 1 is a chart that depicts field strength as a function of distancefor a guided electromagnetic field and a radiated electromagnetic field.

FIG. 2 is a drawing that illustrates a propagation interface with tworegions employed for transmission of a guided surface wave according tovarious embodiments of the present disclosure.

FIG. 3 is a drawing that illustrates a guided surface waveguide probedisposed with respect to a propagation interface of FIG. 2 according tovarious embodiments of the present disclosure.

FIG. 4 is a plot of an example of the magnitudes of close-in and far-outasymptotes of first order Hankel functions according to variousembodiments of the present disclosure.

FIGS. 5A and 5B are drawings that illustrate a complex angle ofincidence of an electric field synthesized by a guided surface waveguideprobe according to various embodiments of the present disclosure.

FIG. 6 is a graphical representation illustrating the effect ofelevation of a charge terminal on the location where the electric fieldof FIG. 5A intersects with the lossy conducting medium at a Brewsterangle according to various embodiments of the present disclosure.

FIG. 7 is a graphical representation of an example of a guided surfacewaveguide probe according to various embodiments of the presentdisclosure.

FIGS. 8A through 8C are graphical representations illustrating examplesof equivalent image plane models of the guided surface waveguide probeof FIGS. 3 and 7 according to various embodiments of the presentdisclosure.

FIGS. 9A and 9B are graphical representations illustrating examples ofsingle-wire transmission line and classic transmission line models ofthe equivalent image plane models of FIGS. 8B and 8C according tovarious embodiments of the present disclosure.

FIG. 10 is a flow chart illustrating an example of adjusting a guidedsurface waveguide probe of FIGS. 3 and 7 to launch a guided surface wavealong the surface of a lossy conducting medium according to variousembodiments of the present disclosure.

FIG. 11 is a plot illustrating an example of the relationship between awave tilt angle and the phase delay of a guided surface waveguide probeof FIGS. 3 and 7 according to various embodiments of the presentdisclosure.

FIG. 12 is a drawing that illustrates an example of a guided surfacewaveguide probe according to various embodiments of the presentdisclosure.

FIG. 13 is a graphical representation illustrating the incidence of asynthesized electric field at a complex Brewster angle to match theguided surface waveguide mode at the Hankel crossover distance accordingto various embodiments of the present disclosure.

FIG. 14 is a graphical representation of an example of a guided surfacewaveguide probe of FIG. 12 according to various embodiments of thepresent disclosure.

FIG. 15A includes plots of an example of the imaginary and real parts ofa phase delay (Φ_(U)) of a charge terminal T₁ of a guided surfacewaveguide probe according to various embodiments of the presentdisclosure.

FIG. 15B is a schematic diagram of the guided surface waveguide probe ofFIG. 14 according to various embodiments of the present disclosure.

FIG. 16 is a drawing that illustrates an example of a guided surfacewaveguide probe according to various embodiments of the presentdisclosure.

FIG. 17 is a graphical representation of an example of a guided surfacewaveguide probe of FIG. 16 according to various embodiments of thepresent disclosure.

FIGS. 18A through 18C depict examples of receiving structures that canbe employed to receive energy transmitted in the form of a guidedsurface wave launched by a guided surface waveguide probe according tothe various embodiments of the present disclosure.

FIG. 18D is a flow chart illustrating an example of adjusting areceiving structure according to various embodiments of the presentdisclosure.

FIG. 19 depicts an example of an additional receiving structure that canbe employed to receive energy transmitted in the form of a guidedsurface wave launched by a guided surface waveguide probe according tothe various embodiments of the present disclosure.

FIGS. 20-20E depict examples of various schematic symbols that are usedaccording to various embodiments of the present disclosure.

FIG. 21 depicts a representation of a hierarchical power distributionsystem including transmission classes and frequency sets according tovarious embodiments of the present disclosure.

FIG. 22 depicts a representation of a transmission spectrum that theguided surface waveguide probes may be tuned into when transmittingelectrical energy in the form of a guided surface wave according tovarious embodiments of the present disclosure.

FIG. 23A depicts another representation of a transmission spectrumaccording to various embodiments of the present disclosure.

FIG. 23B depicts an example of geographic areas that may be served byrespective transmission classes according to various embodiments of thepresent disclosure.

FIG. 24 is a drawing of a networked hierarchical power distributionsystem according to various embodiments of the present disclosure.

FIG. 25 is a flowchart illustrating one example of the transmissioncontrol system according to various embodiments of the presentdisclosure.

FIG. 26 is a flowchart illustrating one example of assigning at leastone discrete frequency within a frequency set to a guided surface wavereceive structure according to various embodiments of the presentdisclosure.

FIG. 27 is a schematic block diagram that provides one exampleillustration of a transmission control system and a receive controlleremployed in the networked hierarchical power distribution system of FIG.24 according to various embodiments of the present disclosure.

DETAILED DESCRIPTION

To begin, some terminology shall be established to provide clarity inthe discussion of concepts to follow. First, as contemplated herein, aformal distinction is drawn between radiated electromagnetic fields andguided electromagnetic fields.

As contemplated herein, a radiated electromagnetic field compriseselectromagnetic energy that is emitted from a source structure in theform of waves that are not bound to a waveguide. For example, a radiatedelectromagnetic field is generally a field that leaves an electricstructure such as an antenna and propagates through the atmosphere orother medium and is not bound to any waveguide structure. Once radiatedelectromagnetic waves leave an electric structure such as an antenna,they continue to propagate in the medium of propagation (such as air)independent of their source until they dissipate regardless of whetherthe source continues to operate. Once electromagnetic waves areradiated, they are not recoverable unless intercepted, and, if notintercepted, the energy inherent in the radiated electromagnetic wavesis lost forever. Electrical structures such as antennas are designed toradiate electromagnetic fields by maximizing the ratio of the radiationresistance to the structure loss resistance. Radiated energy spreads outin space and is lost regardless of whether a receiver is present. Theenergy density of the radiated fields is a function of distance due togeometric spreading. Accordingly, the term “radiate” in all its forms asused herein refers to this form of electromagnetic propagation.

A guided electromagnetic field is a propagating electromagnetic wavewhose energy is concentrated within or near boundaries between mediahaving different electromagnetic properties. In this sense, a guidedelectromagnetic field is one that is bound to a waveguide and may becharacterized as being conveyed by the current flowing in the waveguide.If there is no load to receive and/or dissipate the energy conveyed in aguided electromagnetic wave, then no energy is lost except for thatdissipated in the conductivity of the guiding medium. Stated anotherway, if there is no load for a guided electromagnetic wave, then noenergy is consumed. Thus, a generator or other source generating aguided electromagnetic field does not deliver real power unless aresistive load is present. To this end, such a generator or other sourceessentially runs idle until a load is presented. This is akin to runninga generator to generate a 60 Hertz electromagnetic wave that istransmitted over power lines where there is no electrical load. Itshould be noted that a guided electromagnetic field or wave is theequivalent to what is termed a “transmission line mode.” This contrastswith radiated electromagnetic waves in which real power is supplied atall times in order to generate radiated waves. Unlike radiatedelectromagnetic waves, guided electromagnetic energy does not continueto propagate along a finite length waveguide after the energy source isturned off. Accordingly, the term “guide” in all its forms as usedherein refers to this transmission mode of electromagnetic propagation.

Referring now to FIG. 1, shown is a graph 100 of field strength indecibels (dB) above an arbitrary reference in volts per meter as afunction of distance in kilometers on a log-dB plot to furtherillustrate the distinction between radiated and guided electromagneticfields. The graph 100 of FIG. 1 depicts a guided field strength curve103 that shows the field strength of a guided electromagnetic field as afunction of distance. This guided field strength curve 103 isessentially the same as a transmission line mode. Also, the graph 100 ofFIG. 1 depicts a radiated field strength curve 106 that shows the fieldstrength of a radiated electromagnetic field as a function of distance.

Of interest are the shapes of the curves 103 and 106 for guided wave andfor radiation propagation, respectively. The radiated field strengthcurve 106 falls off geometrically (1/d, where d is distance), which isdepicted as a straight line on the log-log scale. The guided fieldstrength curve 103, on the other hand, has a characteristic exponentialdecay of e^(−αd)/√{square root over (d)} and exhibits a distinctive knee109 on the log-log scale. The guided field strength curve 103 and theradiated field strength curve 106 intersect at point 112, which occursat a crossing distance. At distances less than the crossing distance atintersection point 112, the field strength of a guided electromagneticfield is significantly greater at most locations than the field strengthof a radiated electromagnetic field. At distances greater than thecrossing distance, the opposite is true. Thus, the guided and radiatedfield strength curves 103 and 106 further illustrate the fundamentalpropagation difference between guided and radiated electromagneticfields. For an informal discussion of the difference between guided andradiated electromagnetic fields, reference is made to Milligan, T.,Modem Antenna Design, McGraw-Hill, 1^(st) Edition, 1985, pp. 8-9, whichis incorporated herein by reference in its entirety.

The distinction between radiated and guided electromagnetic waves, madeabove, is readily expressed formally and placed on a rigorous basis.That two such diverse solutions could emerge from one and the samelinear partial differential equation, the wave equation, analyticallyfollows from the boundary conditions imposed on the problem. The Greenfunction for the wave equation, itself, contains the distinction betweenthe nature of radiation and guided waves.

In empty space, the wave equation is a differential operator whoseeigenfunctions possess a continuous spectrum of eigenvalues on thecomplex wave-number plane. This transverse electro-magnetic (TEM) fieldis called the radiation field, and those propagating fields are called“Hertzian waves.” However, in the presence of a conducting boundary, thewave equation plus boundary conditions mathematically lead to a spectralrepresentation of wave-numbers composed of a continuous spectrum plus asum of discrete spectra. To this end, reference is made to Sommerfeld,A., “Uber die Ausbreitung der Wellen in der Drahtlosen Telegraphie,”Annalen der Physik, Vol. 28, 1909, pp. 665-736. Also see Sommerfeld, A.,“Problems of Radio,” published as Chapter 6 in Partial DifferentialEquations in Physics—Lectures on Theoretical Physics: Volume VI,Academic Press, 1949, pp. 236-289, 295-296; Collin, R. E., “HertzianDipole Radiating Over a Lossy Earth or Sea: Some Early and Late 20^(th)Century Controversies,” IEEE Antennas and Propagation Magazine, Vol. 46,No. 2, April 2004, pp. 64-79; and Reich, H. J., Ordnung, P. F, Krauss,H. L., and Skalnik, J. G., Microwave Theory and Techniques, VanNostrand, 1953, pp. 291-293, each of these references being incorporatedherein by reference in its entirety.

The terms “ground wave” and “surface wave” identify two distinctlydifferent physical propagation phenomena. A surface wave arisesanalytically from a distinct pole yielding a discrete component in theplane wave spectrum. See, e.g., “The Excitation of Plane Surface Waves”by Cullen, A. L., (Proceedings of the IEE (British), Vol. 101, Part IV,August 1954, pp. 225-235). In this context, a surface wave is consideredto be a guided surface wave. The surface wave (in the Zenneck-Sommerfeldguided wave sense) is, physically and mathematically, not the same asthe ground wave (in the Weyl-Norton-FCC sense) that is now so familiarfrom radio broadcasting. These two propagation mechanisms arise from theexcitation of different types of eigenvalue spectra (continuum ordiscrete) on the complex plane. The field strength of the guided surfacewave decays exponentially with distance as illustrated by curve 103 ofFIG. 1 (much like propagation in a lossy waveguide) and resemblespropagation in a radial transmission line, as opposed to the classicalHertzian radiation of the ground wave, which propagates spherically,possesses a continuum of eigenvalues, falls off geometrically asillustrated by curve 106 of FIG. 1, and results from branch-cutintegrals. As experimentally demonstrated by C. R. Burrows in “TheSurface Wave in Radio Propagation over Plane Earth” (Proceedings of theIRE, Vol. 25, No. 2, February, 1937, pp. 219-229) and “The Surface Wavein Radio Transmission” (Bell Laboratories Record, Vol. 15, June 1937,pp. 321-324), vertical antennas radiate ground waves but do not launchguided surface waves.

To summarize the above, first, the continuous part of the wave-numbereigenvalue spectrum, corresponding to branch-cut integrals, produces theradiation field, and second, the discrete spectra, and correspondingresidue sum arising from the poles enclosed by the contour ofintegration, result in non-TEM traveling surface waves that areexponentially damped in the direction transverse to the propagation.Such surface waves are guided transmission line modes. For furtherexplanation, reference is made to Friedman, B., Principles andTechniques of Applied Mathematics, Wiley, 1956, pp. pp. 214, 283-286,290, 298-300.

In free space, antennas excite the continuum eigenvalues of the waveequation, which is a radiation field, where the outwardly propagating RFenergy with E_(z) and H_(φ) in-phase is lost forever. On the other hand,waveguide probes excite discrete eigenvalues, which results intransmission line propagation. See Collin, R. E., Field Theory of GuidedWaves, McGraw-Hill, 1960, pp. 453, 474-477. While such theoreticalanalyses have held out the hypothetical possibility of launching opensurface guided waves over planar or spherical surfaces of lossy,homogeneous media, for more than a century no known structures in theengineering arts have existed for accomplishing this with any practicalefficiency. Unfortunately, since it emerged in the early 1900's, thetheoretical analysis set forth above has essentially remained a theoryand there have been no known structures for practically accomplishingthe launching of open surface guided waves over planar or sphericalsurfaces of lossy, homogeneous media.

According to the various embodiments of the present disclosure, variousguided surface waveguide probes are described that are configured toexcite electric fields that couple into a guided surface waveguide modealong the surface of a lossy conducting medium. Such guidedelectromagnetic fields are substantially mode-matched in magnitude andphase to a guided surface wave mode on the surface of the lossyconducting medium. Such a guided surface wave mode can also be termed aZenneck waveguide mode. By virtue of the fact that the resultant fieldsexcited by the guided surface waveguide probes described herein aresubstantially mode-matched to a guided surface waveguide mode on thesurface of the lossy conducting medium, a guided electromagnetic fieldin the form of a guided surface wave is launched along the surface ofthe lossy conducting medium. According to one embodiment, the lossyconducting medium comprises a terrestrial medium such as the Earth.

Referring to FIG. 2, shown is a propagation interface that provides foran examination of the boundary value solutions to Maxwell's equationsderived in 1907 by Jonathan Zenneck as set forth in his paper Zenneck,J., “On the Propagation of Plane Electromagnetic Waves Along a FlatConducting Surface and their Relation to Wireless Telegraphy,” Annalender Physik, Serial 4, Vol. 23, Sep. 20, 1907, pp. 846-866. FIG. 2depicts cylindrical coordinates for radially propagating waves along theinterface between a lossy conducting medium specified as Region 1 and aninsulator specified as Region 2. Region 1 can comprise, for example, anylossy conducting medium. In one example, such a lossy conducting mediumcan comprise a terrestrial medium such as the Earth or other medium.Region 2 is a second medium that shares a boundary interface with Region1 and has different constitutive parameters relative to Region 1. Region2 can comprise, for example, any insulator such as the atmosphere orother medium. The reflection coefficient for such a boundary interfacegoes to zero only for incidence at a complex Brewster angle. SeeStratton, J. A., Electromagnetic Theory, McGraw-Hill, 1941, p. 516.

According to various embodiments, the present disclosure sets forthvarious guided surface waveguide probes that generate electromagneticfields that are substantially mode-matched to a guided surface waveguidemode on the surface of the lossy conducting medium comprising Region 1.According to various embodiments, such electromagnetic fieldssubstantially synthesize a wave front incident at a complex Brewsterangle of the lossy conducting medium that can result in zero reflection.

To explain further, in Region 2, where an e^(jωt) field variation isassumed and where ρ≠0 and z≥0 (with z being the vertical coordinatenormal to the surface of Region 1, and ρ being the radial dimension incylindrical coordinates), Zenneck's closed-form exact solution ofMaxwell's equations satisfying the boundary conditions along theinterface are expressed by the following electric field and magneticfield components:

$\begin{matrix}{{H_{2\phi} = {A\; e^{{- u_{2}}z}{H_{1}^{(2)}\left( {{- j}\;\gamma\;\rho} \right)}}},} & (1) \\{{E_{2\;\rho} = {{A\left( \frac{u_{2}}{j\;\omega\; ɛ_{0}} \right)}e^{{- u_{2}}z}{H_{1}^{(2)}\left( {{- j}\;\gamma\;\rho} \right)}}},{and}} & (2) \\{E_{2z} = {{A\left( \frac{- \gamma}{\omega\; ɛ_{0}} \right)}e^{{- u_{2}}z}{{H_{0}^{(2)}\left( {{- j}\;\gamma\;\rho} \right)}.}}} & (3)\end{matrix}$

In Region 1, where the e^(jωt) field variation is assumed and where ρ≠0and z≤0, Zenneck's closed-form exact solution of Maxwell's equationssatisfying the boundary conditions along the interface is expressed bythe following electric field and magnetic field components:

$\begin{matrix}{{H_{1\phi} = {A\; e^{u_{1}z}{H_{1}^{(2)}\left( {{- j}\;\gamma\;\rho} \right)}}},} & (4) \\{{E_{1\;\rho} = {{A\left( \frac{- u_{1}}{\sigma_{1} + {j\;\omega\; ɛ_{1}}} \right)}e^{u_{1}z}{H_{1}^{(2)}\left( {{- j}\;\gamma\;\rho} \right)}}},{and}} & (5) \\{E_{1z} = {{A\left( \frac{{- j}\;\gamma}{\sigma_{1} + {j\;\omega\; ɛ_{1}}} \right)}e^{u_{1}z}{{H_{0}^{(2)}\left( {{- j}\;\gamma\;\rho} \right)}.}}} & (6)\end{matrix}$

In these expressions, z is the vertical coordinate normal to the surfaceof Region 1 and ρ is the radial coordinate, H_(n) ⁽²⁾(−jγρ) is a complexargument Hankel function of the second kind and order n, u₁ is thepropagation constant in the positive vertical (z) direction in Region 1,u₂ is the propagation constant in the vertical (z) direction in Region2, σ₁ is the conductivity of Region 1, ω is equal to 2πf, where f is afrequency of excitation, ε_(o) is the permittivity of free space, ε₁ isthe permittivity of Region 1, A is a source constant imposed by thesource, and γ is a surface wave radial propagation constant.

The propagation constants in the +z directions are determined byseparating the wave equation above and below the interface betweenRegions 1 and 2, and imposing the boundary conditions. This exercisegives, in Region 2,

$\begin{matrix}{u_{2} = \frac{{- j}\; k_{o}}{\sqrt{1 + \left( {ɛ_{r} - {j\; x}} \right)}}} & (7)\end{matrix}$and gives, in Region 1,u ₁ =−u ₂(ε_(r) −jx).  (8)The radial propagation constant γ is given by

$\begin{matrix}{{\gamma = {{j\sqrt{k_{o}^{2} + u_{2}^{2}}} = {j\frac{k_{o}n}{\sqrt{1 + n^{2}}}}}},} & (9)\end{matrix}$which is a complex expression where n is the complex index of refractiongiven byn=√{square root over (ε_(r) −jx)}.  (10)In all of the above Equations,

$\begin{matrix}{{x = \frac{\sigma_{1}}{\omega\; ɛ_{o}}},{and}} & (11) \\{{k_{o} = {{\omega\sqrt{\mu_{o}ɛ_{o}}} = \frac{\lambda_{o}}{2\pi}}},} & (12)\end{matrix}$where ε_(r) comprises the relative permittivity of Region 1, σ₁ is theconductivity of Region 1, ε_(o) is the permittivity of free space, andμ_(o) comprises the permeability of free space. Thus, the generatedsurface wave propagates parallel to the interface and exponentiallydecays vertical to it. This is known as evanescence.

Thus, Equations (1)-(3) can be considered to be acylindrically-symmetric, radially-propagating waveguide mode. SeeBarlow, H. M., and Brown, J., Radio Surface Waves, Oxford UniversityPress, 1962, pp. 10-12, 29-33. The present disclosure details structuresthat excite this “open boundary” waveguide mode. Specifically, accordingto various embodiments, a guided surface waveguide probe is providedwith a charge terminal of appropriate size that is fed with voltageand/or current and is positioned relative to the boundary interfacebetween Region 2 and Region 1. This may be better understood withreference to FIG. 3, which shows an example of a guided surfacewaveguide probe 200 a that includes a charge terminal T₁ elevated abovea lossy conducting medium 203 (e.g., the Earth) along a vertical axis zthat is normal to a plane presented by the lossy conducting medium 203.The lossy conducting medium 203 makes up Region 1, and a second medium206 makes up Region 2 and shares a boundary interface with the lossyconducting medium 203.

According to one embodiment, the lossy conducting medium 203 cancomprise a terrestrial medium such as the planet Earth. To this end,such a terrestrial medium comprises all structures or formationsincluded thereon whether natural or man-made. For example, such aterrestrial medium can comprise natural elements such as rock, soil,sand, fresh water, sea water, trees, vegetation, and all other naturalelements that make up our planet. In addition, such a terrestrial mediumcan comprise man-made elements such as concrete, asphalt, buildingmaterials, and other man-made materials. In other embodiments, the lossyconducting medium 203 can comprise some medium other than the Earth,whether naturally occurring or man-made. In other embodiments, the lossyconducting medium 203 can comprise other media such as man-made surfacesand structures such as automobiles, aircraft, man-made materials (suchas plywood, plastic sheeting, or other materials) or other media.

In the case where the lossy conducting medium 203 comprises aterrestrial medium or Earth, the second medium 206 can comprise theatmosphere above the ground. As such, the atmosphere can be termed an“atmospheric medium” that comprises air and other elements that make upthe atmosphere of the Earth. In addition, it is possible that the secondmedium 206 can comprise other media relative to the lossy conductingmedium 203.

The guided surface waveguide probe 200 a includes a feed network 209that couples an excitation source 212 to the charge terminal T₁ via,e.g., a vertical feed line conductor. According to various embodiments,a charge Q₁ is imposed on the charge terminal T₁ to synthesize anelectric field based upon the voltage applied to terminal T₁ at anygiven instant. Depending on the angle of incidence (θ_(i)) of theelectric field (E), it is possible to substantially mode-match theelectric field to a guided surface waveguide mode on the surface of thelossy conducting medium 203 comprising Region 1.

By considering the Zenneck closed-form solutions of Equations (1)-(6),the Leontovich impedance boundary condition between Region 1 and Region2 can be stated as{circumflex over (z)}×

₂(ρ,φ,0)=

_(S),  (13)where {circumflex over (z)} is a unit normal in the positive vertical(+z) direction and

₂ is the magnetic field strength in Region 2 expressed by Equation (1)above. Equation (13) implies that the electric and magnetic fieldsspecified in Equations (1)-(3) may result in a radial surface currentdensity along the boundary interface, where the radial surface currentdensity can be specified byJ _(ρ)(ρ′)=−AH ₁ ⁽²⁾(−jγρ′)  (14)where A is a constant. Further, it should be noted that close-in to theguided surface waveguide probe 200 (for ρ<<λ), Equation (14) above hasthe behavior

$\begin{matrix}{{J_{close}\left( \rho^{\prime} \right)} = {\frac{- {A\left( {j\; 2} \right)}}{\pi\left( {{- j}\;\gamma\;\rho^{\prime}} \right)} = {{- H_{\phi}} = {- {\frac{I_{o}}{2\pi\;\rho^{\prime}}.}}}}} & (15)\end{matrix}$The negative sign means that when source current (I_(o)) flowsvertically upward as illustrated in FIG. 3, the “close-in” groundcurrent flows radially inward. By field matching on H_(φ) “close-in,” itcan be determined that

$\begin{matrix}{A = {{- \frac{I_{o}\gamma}{4}} = {- \frac{\omega\; q_{1}\gamma}{4}}}} & (16)\end{matrix}$where q₁=C₁V₁, in Equations (1)-(6) and (14). Therefore, the radialsurface current density of Equation (14) can be restated as

$\begin{matrix}{{J_{\rho}\left( \rho^{\prime} \right)} = {\frac{I_{o}\gamma}{4}{{H_{1}^{(2)}\left( {{- j}\;\gamma\;\rho^{\prime}} \right)}.}}} & (17)\end{matrix}$The fields expressed by Equations (1)-(6) and (17) have the nature of atransmission line mode bound to a lossy interface, not radiation fieldsthat are associated with groundwave propagation. See Barlow, H. M. andBrown, J., Radio Surface Waves, Oxford University Press, 1962, pp. 1-5.

At this point, a review of the nature of the Hankel functions used inEquations (1)-(6) and (17) is provided for these solutions of the waveequation. One might observe that the Hankel functions of the first andsecond kind and order n are defined as complex combinations of thestandard Bessel functions of the first and second kindsH _(n) ⁽¹⁾(x)=J _(n)(x)+jN _(n)(x), and  (18)H _(n) ⁽²⁾(x)=J _(n)(x)−jN _(n)(x),  (19)These functions represent cylindrical waves propagating radially inward(H_(n) ⁽¹⁾) and outward (H_(n) ⁽²⁾), respectively. The definition isanalogous to the relationship e^(±jx)=cos x±j sin x. See, for example,Harrington, R. F., Time-Harmonic Fields, McGraw-Hill, 1961, pp. 460-463.

That H_(n) ⁽²⁾(k_(ρ)ρ) is an outgoing wave can be recognized from itslarge argument asymptotic behavior that is obtained directly from theseries definitions of J_(n)(x) and N_(n)(x). Far-out from the guidedsurface waveguide probe:

$\begin{matrix}{{{{H_{n}^{(2)}(x)}\underset{\;{x\rightarrow\infty}\mspace{14mu}}{\rightarrow}{\sqrt{\frac{2j}{\pi\; x}}j^{n}e^{{- j}\; x}}} = {\sqrt{\frac{2}{\pi\; x}}j^{n}e^{- {j{({x - \frac{\pi}{4}})}}}}},} & \left( {20a} \right)\end{matrix}$which, when multiplied by e^(jωt), is an outward propagating cylindricalwave of the form e^(j(ωt−kρ)) with a 1/√{square root over (ρ)} spatialvariation. The first order (n=1) solution can be determined fromEquation (20a) to be

$\begin{matrix}{{{H_{1}^{(2)}(x)}\underset{\;{x\rightarrow\infty}\mspace{20mu}}{\rightarrow}{j\sqrt{\frac{2j}{\pi\; x}}e^{{- j}\; x}}} = {\sqrt{\frac{2}{\pi\; x}}{e^{- {j{({x - \frac{\pi}{2} - \frac{\pi}{4}})}}}.}}} & \left( {20b} \right)\end{matrix}$Close-in to the guided surface waveguide probe (for ρ<<λ), the Hankelfunction of first order and the second kind behaves as

$\begin{matrix}{{H_{1}^{(2)}(x)}\underset{\;{x\rightarrow 0}\mspace{14mu}}{\rightarrow}{\frac{2j}{\pi\; x}.}} & (21)\end{matrix}$Note that these asymptotic expressions are complex quantities. When x isa real quantity, Equations (20b) and (21) differ in phase by √{squareroot over (j)}, which corresponds to an extra phase advance or “phaseboost” of 45° or, equivalently, λ/8. The close-in and far-out asymptotesof the first order Hankel function of the second kind have a Hankel“crossover” or transition point where they are of equal magnitude at adistance of ρ=R_(x).

Thus, beyond the Hankel crossover point the “far out” representationpredominates over the “close-in” representation of the Hankel function.The distance to the Hankel crossover point (or Hankel crossoverdistance) can be found by equating Equations (20b) and (21) for −jγp,and solving for R_(x). With x=σ/ωε_(o), it can be seen that the far-outand close-in Hankel function asymptotes are frequency dependent, withthe Hankel crossover point moving out as the frequency is lowered. Itshould also be noted that the Hankel function asymptotes may also varyas the conductivity (σ) of the lossy conducting medium changes. Forexample, the conductivity of the soil can vary with changes in weatherconditions.

Referring to FIG. 4, shown is an example of a plot of the magnitudes ofthe first order Hankel functions of Equations (20b) and (21) for aRegion 1 conductivity of σ=0.010 mhos/m and relative permittivityε_(r)=15, at an operating frequency of 1850 kHz. Curve 115 is themagnitude of the far-out asymptote of Equation (20b) and curve 118 isthe magnitude of the close-in asymptote of Equation (21), with theHankel crossover point 121 occurring at a distance of R_(x)=54 feet.While the magnitudes are equal, a phase offset exists between the twoasymptotes at the Hankel crossover point 121. It can also be seen thatthe Hankel crossover distance is much less than a wavelength of theoperation frequency.

Considering the electric field components given by Equations (2) and (3)of the Zenneck closed-form solution in Region 2, it can be seen that theratio of E_(z) and E_(ρ) asymptotically passes to

$\begin{matrix}{{\frac{E_{z}}{E_{\rho}} = {{{\left( \frac{{- j}\;\gamma}{u_{2}} \right)\frac{H_{0}^{(2)}\left( {{- j}\;\gamma\;\rho} \right)}{H_{1}^{(2)}\left( {{- j}\;\gamma\;\rho} \right)}}\underset{\;{\rho\rightarrow\infty}\mspace{14mu}}{\rightarrow}\sqrt{ɛ_{r} - {j\frac{\sigma}{\omega\; ɛ_{o}}}}} = {n = {\tan\;\theta_{i}}}}},} & (22)\end{matrix}$where n is the complex index of refraction of Equation (10) and θ_(i) isthe angle of incidence of the electric field. In addition, the verticalcomponent of the mode-matched electric field of Equation (3)asymptotically passes to

$\begin{matrix}{{E_{2z}\underset{\;{\rho\rightarrow\infty}\mspace{14mu}}{\rightarrow}{\left( \frac{q_{free}}{ɛ_{o}} \right)\sqrt{\frac{\gamma^{3}}{8\pi}}e^{{- u_{2}}z}\frac{e^{- {j{({{\gamma\;\rho} - {\pi/4}})}}}}{\sqrt{\rho}}}},} & (23)\end{matrix}$which is linearly proportional to free charge on the isolated componentof the elevated charge terminal's capacitance at the terminal voltage,q_(free)=C_(free)×V_(T).

For example, the height H₁ of the elevated charge terminal T₁ in FIG. 3affects the amount of free charge on the charge terminal T₁. When thecharge terminal T₁ is near the ground plane of Region 1, most of thecharge Q₁ on the terminal is “bound.” As the charge terminal T₁ iselevated, the bound charge is lessened until the charge terminal T₁reaches a height at which substantially all of the isolated charge isfree.

The advantage of an increased capacitive elevation for the chargeterminal T₁ is that the charge on the elevated charge terminal T₁ isfurther removed from the ground plane, resulting in an increased amountof free charge q_(free) to couple energy into the guided surfacewaveguide mode. As the charge terminal T₁ is moved away from the groundplane, the charge distribution becomes more uniformly distributed aboutthe surface of the terminal. The amount of free charge is related to theself-capacitance of the charge terminal T₁.

For example, the capacitance of a spherical terminal can be expressed asa function of physical height above the ground plane. The capacitance ofa sphere at a physical height of h above a perfect ground is given byC _(elevated sphere)=4πε_(o) a(1+M+M ² +M ³+2M ⁴+3M ⁵+ . . . ),  (24)where the diameter of the sphere is 2a, and where M=a/2h with h beingthe height of the spherical terminal. As can be seen, an increase in theterminal height h reduces the capacitance C of the charge terminal. Itcan be shown that for elevations of the charge terminal T₁ that are at aheight of about four times the diameter (4D=8a) or greater, the chargedistribution is approximately uniform about the spherical terminal,which can improve the coupling into the guided surface waveguide mode.

In the case of a sufficiently isolated terminal, the self-capacitance ofa conductive sphere can be approximated by C=4πε_(o)a, where a is theradius of the sphere in meters, and the self-capacitance of a disk canbe approximated by C=8ε_(o)a, where a is the radius of the disk inmeters. The charge terminal T₁ can include any shape such as a sphere, adisk, a cylinder, a cone, a torus, a hood, one or more rings, or anyother randomized shape or combination of shapes. An equivalent sphericaldiameter can be determined and used for positioning of the chargeterminal T₁.

This may be further understood with reference to the example of FIG. 3,where the charge terminal T₁ is elevated at a physical height ofh_(p)=H₁ above the lossy conducting medium 203. To reduce the effects ofthe “bound” charge, the charge terminal T₁ can be positioned at aphysical height that is at least four times the spherical diameter (orequivalent spherical diameter) of the charge terminal T₁ to reduce thebounded charge effects.

Referring next to FIG. 5A, shown is a ray optics interpretation of theelectric field produced by the elevated charge Q₁ on charge terminal T₁of FIG. 3. As in optics, minimizing the reflection of the incidentelectric field can improve and/or maximize the energy coupled into theguided surface waveguide mode of the lossy conducting medium 203. For anelectric field (E_(∥)) that is polarized parallel to the plane ofincidence (not the boundary interface), the amount of reflection of theincident electric field may be determined using the Fresnel reflectioncoefficient, which can be expressed as

$\begin{matrix}{{{\Gamma_{}\left( \theta_{i} \right)} = {\frac{E_{{,R}}}{E_{{,i}}} = \frac{\sqrt{\left( {ɛ_{r} - {j\; x}} \right) - {\sin^{2}\theta_{i}}} - {\left( {ɛ_{r} - {j\; x}} \right)\cos\;\theta_{i}}}{\sqrt{\left( {ɛ_{r} - {j\; x}} \right) - {\sin^{2}\theta_{i}}} + {\left( {ɛ_{r} - {j\; x}} \right)\cos\;\theta_{i}}}}},} & (25)\end{matrix}$where θ_(i) is the conventional angle of incidence measured with respectto the surface normal.

In the example of FIG. 5A, the ray optic interpretation shows theincident field polarized parallel to the plane of incidence having anangle of incidence of θ_(i) which is measured with respect to thesurface normal ({circumflex over (z)}). There will be no reflection ofthe incident electric field when Γ_(∥)(θ_(i))=0 and thus the incidentelectric field will be completely coupled into a guided surfacewaveguide mode along the surface of the lossy conducting medium 203. Itcan be seen that the numerator of Equation (25) goes to zero when theangle of incidence isθ_(i)=arctan(√{square root over (εr−jx)})=θ_(i,B),  (26)where x=σ/ωε_(o). This complex angle of incidence (θ_(i,B)) is referredto as the Brewster angle. Referring back to Equation (22), it can beseen that the same complex Brewster angle (θ_(i,B)) relationship ispresent in both Equations (22) and (26).

As illustrated in FIG. 5A, the electric field vector E can be depictedas an incoming non-uniform plane wave, polarized parallel to the planeof incidence. The electric field vector E can be created fromindependent horizontal and vertical components as

(θ_(i))=E _(ρ) {circumflex over (ρ)}+E _(z) {circumflex over (z)}.  (27)Geometrically, the illustration in FIG. 5A suggests that the electricfield vector E can be given by

$\begin{matrix}{{{E_{\rho}\left( {\rho,z} \right)} = {{E\left( {\rho,z} \right)}\cos\;\theta_{i}}},{and}} & \left( {28\; a} \right) \\{{{E_{z}\left( {\rho,z} \right)} = {{{E\left( {\rho,z} \right)}{\cos\left( {\frac{\pi}{2} - \theta_{i}} \right)}} = {{E\left( {\rho,z} \right)}\sin\;\theta_{i}}}},} & \left( {28\; b} \right)\end{matrix}$which means that the field ratio is

$\begin{matrix}{\frac{E_{\rho}}{E_{z}} = {\frac{1}{\tan\;\theta_{i}} = {\tan\;{\psi_{i}.}}}} & (29)\end{matrix}$

A generalized parameter W, called “wave tilt,” is noted herein as theratio of the horizontal electric field component to the verticalelectric field component given by

$\begin{matrix}{{W = {\frac{E_{\rho}}{E_{z}} = {{W}e^{j\;\Psi}}}},{or}} & \left( {30\; a} \right) \\{{\frac{1}{W} = {\frac{E_{z}}{E_{\rho}} = {{\tan\;\theta_{i}} = {\frac{1}{W}e^{{- j}\;\Psi}}}}},} & \left( {30\; b} \right)\end{matrix}$which is complex and has both magnitude and phase. For anelectromagnetic wave in Region 2, the wave tilt angle (ψ) is equal tothe angle between the normal of the wave-front at the boundary interfacewith Region 1 and the tangent to the boundary interface. This may beeasier to see in FIG. 5B, which illustrates equi-phase surfaces of anelectromagnetic wave and their normals for a radial cylindrical guidedsurface wave. At the boundary interface (z=0) with a perfect conductor,the wave-front normal is parallel to the tangent of the boundaryinterface, resulting in W=0. However, in the case of a lossy dielectric,a wave tilt W exists because the wave-front normal is not parallel withthe tangent of the boundary interface at z=0.

Applying Equation (30b) to a guided surface wave gives

$\begin{matrix}{{\tan\;\theta_{i,B}} = {\frac{E_{z}}{E_{\rho}} = {\frac{u_{2}}{\gamma} = {\sqrt{ɛ_{r} - {j\; x}} = {n = {\frac{1}{W} = {\frac{1}{W}{e^{{- j}\;\Psi}.}}}}}}}} & (31)\end{matrix}$

With the angle of incidence equal to the complex Brewster angle(θ_(i,B)), the Fresnel reflection coefficient of Equation (25) vanishes,as shown by

$\begin{matrix}{{\Gamma_{}\left( \theta_{i,B} \right)} = {\left. \frac{\sqrt{\left( {ɛ_{r} - {j\; x}} \right) - {\sin^{2}\theta_{i}}} - {\left( {ɛ_{r} - {j\; x}} \right)\cos\;\theta_{i}}}{\sqrt{\left( {ɛ_{r} - {j\; x}} \right) - {\sin^{2}\theta_{i}}} + {\left( {ɛ_{r} - {j\; x}} \right)\cos\;\theta_{i}}} \right|_{\theta_{i} = \theta_{i,B}} = 0.}} & (32)\end{matrix}$By adjusting the complex field ratio of Equation (22), an incident fieldcan be synthesized to be incident at a complex angle at which thereflection is reduced or eliminated. Establishing this ratio asn=√{square root over (ε_(r)−jx)} results in the synthesized electricfield being incident at the complex Brewster angle, making thereflections vanish.

The concept of an electrical effective height can provide furtherinsight into synthesizing an electric field with a complex angle ofincidence with a guided surface waveguide probe 200. The electricaleffective height (h_(eff)) has been defined as

$\begin{matrix}{h_{eff} = {\frac{1}{I_{0}}{\int_{0}^{h_{p}}{{I(z)}d\; z}}}} & (33)\end{matrix}$for a monopole with a physical height (or length) of h_(p). Since theexpression depends upon the magnitude and phase of the sourcedistribution along the structure, the effective height (or length) iscomplex in general. The integration of the distributed current I(z) ofthe structure is performed over the physical height of the structure(h_(p)), and normalized to the ground current (I₀) flowing upwardthrough the base (or input) of the structure. The distributed currentalong the structure can be expressed byI(z)=I _(C) cos(β₀ z),  (34)where β₀ is the propagation factor for current propagating on thestructure. In the example of FIG. 3, I_(C) is the current that isdistributed along the vertical structure of the guided surface waveguideprobe 200 a.

For example, consider a feed network 209 that includes a low loss coil(e.g., a helical coil) at the bottom of the structure and a verticalfeed line conductor connected between the coil and the charge terminalT₁. The phase delay due to the coil (or helical delay line) isθ_(c)=β_(p)l_(C), with a physical length of l_(C) and a propagationfactor of

$\begin{matrix}{{\beta_{p} = {\frac{2\pi}{\lambda_{p}} = \frac{2\pi}{V_{f}\lambda_{0}}}},} & (35)\end{matrix}$where V_(f) is the velocity factor on the structure, λ₀ is thewavelength at the supplied frequency, and λ_(p) is the propagationwavelength resulting from the velocity factor V_(f). The phase delay ismeasured relative to the ground (stake) current I₀.

In addition, the spatial phase delay along the length l_(w) of thevertical feed line conductor can be given by θ_(y)=β_(w)l_(w) whereβ_(w) is the propagation phase constant for the vertical feed lineconductor. In some implementations, the spatial phase delay may beapproximated by θ_(y)=β_(w)h_(p), since the difference between thephysical height h_(p) of the guided surface waveguide probe 200 a andthe vertical feed line conductor length 1, is much less than awavelength at the supplied frequency (λ₀). As a result, the total phasedelay through the coil and vertical feed line conductor isΦ=θ_(c)+θ_(y), and the current fed to the top of the coil from thebottom of the physical structure isI _(C)(θ_(c)+θ_(y))=I ₀ e ^(jΦ),  (36)with the total phase delay Φ measured relative to the ground (stake)current I₀. Consequently, the electrical effective height of a guidedsurface waveguide probe 200 can be approximated by

$\begin{matrix}{{h_{eff} = {{\frac{1}{I_{0}}{\int_{0}^{h_{p}}{I_{0}e^{j\;\Phi}{\cos\left( {\beta_{0}z} \right)}d\; z}}} \cong {h_{p}e^{j\;\Phi}}}},} & (37)\end{matrix}$for the case where the physical height h_(p)<<λ₀. The complex effectiveheight of a monopole, h_(eff)=h_(p) at an angle (or phase shift) of Φ,may be adjusted to cause the source fields to match a guided surfacewaveguide mode and cause a guided surface wave to be launched on thelossy conducting medium 203.

In the example of FIG. 5A, ray optics are used to illustrate the complexangle trigonometry of the incident electric field (E) having a complexBrewster angle of incidence (θ_(i,B)) at the Hankel crossover distance(R_(x)) 121. Recall from Equation (26) that, for a lossy conductingmedium, the Brewster angle is complex and specified by

$\begin{matrix}{{\tan\;\theta_{i,B}} = {\sqrt{ɛ_{r} - {j\frac{\sigma}{{\omega ɛ}_{o}}}} = {n.}}} & (38)\end{matrix}$Electrically, the geometric parameters are related by the electricaleffective height (h_(eff)) of the charge terminal T₁ byR _(x) tan ψ_(i,B) =R _(x) ×W=h _(eff) =h _(eff) =h _(p) ^(jΦ),  (39)where ψ_(i,B)=(π/2)−θ_(i,B) is the Brewster angle measured from thesurface of the lossy conducting medium. To couple into the guidedsurface waveguide mode, the wave tilt of the electric field at theHankel crossover distance can be expressed as the ratio of theelectrical effective height and the Hankel crossover distance

$\begin{matrix}{\frac{h_{eff}}{R_{x}} = {{\tan\;\psi_{i,B}} = {W_{Rx}.}}} & (40)\end{matrix}$Since both the physical height (h_(p)) and the Hankel crossover distance(R_(x)) are real quantities, the angle (Ψ) of the desired guided surfacewave tilt at the Hankel crossover distance (R_(x)) is equal to the phase(Φ) of the complex effective height (h_(eff)). This implies that byvarying the phase at the supply point of the coil, and thus the phaseshift in Equation (37), the phase, Φ, of the complex effective heightcan be manipulated to match the angle of the wave tilt, Ψ, of the guidedsurface waveguide mode at the Hankel crossover point 121: Φ=Ψ.

In FIG. 5A, a right triangle is depicted having an adjacent side oflength R_(x) along the lossy conducting medium surface and a complexBrewster angle ψ_(i,B) measured between a ray 124 extending between theHankel crossover point 121 at R_(x) and the center of the chargeterminal T₁, and the lossy conducting medium surface 127 between theHankel crossover point 121 and the charge terminal T₁. With the chargeterminal T₁ positioned at physical height h_(p) and excited with acharge having the appropriate phase delay Φ, the resulting electricfield is incident with the lossy conducting medium boundary interface atthe Hankel crossover distance R_(x), and at the Brewster angle. Underthese conditions, the guided surface waveguide mode can be excitedwithout reflection or substantially negligible reflection.

If the physical height of the charge terminal T₁ is decreased withoutchanging the phase shift Φ of the effective height (h_(eff)), theresulting electric field intersects the lossy conducting medium 203 atthe Brewster angle at a reduced distance from the guided surfacewaveguide probe 200. FIG. 6 graphically illustrates the effect ofdecreasing the physical height of the charge terminal T₁ on the distancewhere the electric field is incident at the Brewster angle. As theheight is decreased from h₃ through h₂ to h₁, the point where theelectric field intersects with the lossy conducting medium (e.g., theEarth) at the Brewster angle moves closer to the charge terminalposition. However, as Equation (39) indicates, the height H₁ (FIG. 3) ofthe charge terminal T₁ should be at or higher than the physical height(h_(p)) in order to excite the far-out component of the Hankel function.With the charge terminal T₁ positioned at or above the effective height(h_(eff)), the lossy conducting medium 203 can be illuminated at theBrewster angle of incidence (ψ_(i,B)=(π/2)−θ_(i,B)) at or beyond theHankel crossover distance (R_(x)) 121 as illustrated in FIG. 5A. Toreduce or minimize the bound charge on the charge terminal T₁, theheight should be at least four times the spherical diameter (orequivalent spherical diameter) of the charge terminal T₁ as mentionedabove.

A guided surface waveguide probe 200 can be configured to establish anelectric field having a wave tilt that corresponds to a waveilluminating the surface of the lossy conducting medium 203 at a complexBrewster angle, thereby exciting radial surface currents bysubstantially mode-matching to a guided surface wave mode at (or beyond)the Hankel crossover point 121 at R_(x).

Referring to FIG. 7, shown is a graphical representation of an exampleof a guided surface waveguide probe 200 b that includes a chargeterminal T₁. An AC source 212 acts as the excitation source for thecharge terminal T₁, which is coupled to the guided surface waveguideprobe 200 b through a feed network 209 (FIG. 3) comprising a coil 215such as, e.g., a helical coil. In other implementations, the AC source212 can be inductively coupled to the coil 215 through a primary coil.In some embodiments, an impedance matching network may be included toimprove and/or maximize coupling of the AC source 212 to the coil 215.

As shown in FIG. 7, the guided surface waveguide probe 200 b can includethe upper charge terminal T₁ (e.g., a sphere at height h_(p)) that ispositioned along a vertical axis z that is substantially normal to theplane presented by the lossy conducting medium 203. A second medium 206is located above the lossy conducting medium 203. The charge terminal T₁has a self-capacitance C_(T). During operation, charge Q₁ is imposed onthe terminal T₁ depending on the voltage applied to the terminal T₁ atany given instant.

In the example of FIG. 7, the coil 215 is coupled to a ground stake 218at a first end and to the charge terminal T₁ via a vertical feed lineconductor 221. In some implementations, the coil connection to thecharge terminal T₁ can be adjusted using a tap 224 of the coil 215 asshown in FIG. 7. The coil 215 can be energized at an operating frequencyby the AC source 212 through a tap 227 at a lower portion of the coil215. In other implementations, the AC source 212 can be inductivelycoupled to the coil 215 through a primary coil.

The construction and adjustment of the guided surface waveguide probe200 is based upon various operating conditions, such as the transmissionfrequency, conditions of the lossy conducting medium (e.g., soilconductivity a and relative permittivity ε_(r)), and size of the chargeterminal T₁. The index of refraction can be calculated from Equations(10) and (11) asn=√{square root over (ε_(r) −jx)},  (41)where x=σ/ωε_(o) with ω=2πf. The conductivity a and relativepermittivity ε_(r) can be determined through test measurements of thelossy conducting medium 203. The complex Brewster angle (θ_(i,B))measured from the surface normal can also be determined from Equation(26) asθ_(i,B)=arctan(√{square root over (ε_(r) −jx)}),  (42)or measured from the surface as shown in FIG. 5A as

$\begin{matrix}{\psi_{i,B} = {\frac{\pi}{2} - {\theta_{i,B}.}}} & (43)\end{matrix}$The wave tilt at the Hankel crossover distance (W_(Rx)) can also befound using Equation (40).

The Hankel crossover distance can also be found by equating themagnitudes of Equations (20b) and (21) for −jγp, and solving for R_(x)as illustrated by FIG. 4. The electrical effective height can then bedetermined from Equation (39) using the Hankel crossover distance andthe complex Brewster angle ash _(eff) =h _(p) e ^(jΦ) =R _(x) tan ψ_(i,B).  (44)As can be seen from Equation (44), the complex effective height(h_(eff)) includes a magnitude that is associated with the physicalheight (h_(p)) of the charge terminal T₁ and a phase delay (Φ) that isto be associated with the angle (Ψ) of the wave tilt at the Hankelcrossover distance (R_(x)). With these variables and the selected chargeterminal T₁ configuration, it is possible to determine the configurationof a guided surface waveguide probe 200.

With the charge terminal T₁ positioned at or above the physical height(h_(p)), the feed network 209 (FIG. 3) and/or the vertical feed lineconnecting the feed network to the charge terminal T₁ can be adjusted tomatch the phase (Φ) of the charge Q₁ on the charge terminal T₁ to theangle (Ψ) of the wave tilt (W). The size of the charge terminal T₁ canbe chosen to provide a sufficiently large surface for the charge Q₁imposed on the terminals. In general, it is desirable to make the chargeterminal T₁ as large as practical. The size of the charge terminal T₁should be large enough to avoid ionization of the surrounding air, whichcan result in electrical discharge or sparking around the chargeterminal.

The phase delay θ_(c) of a helically-wound coil can be determined fromMaxwell's equations as has been discussed by Corum, K. L. and J. F.Corum, “RF Coils, Helical Resonators and Voltage Magnification byCoherent Spatial Modes,” Microwave Review, Vol. 7, No. 2, September2001, pp. 36-45., which is incorporated herein by reference in itsentirety. For a helical coil with H/D>1, the ratio of the velocity ofpropagation (ν) of a wave along the coil's longitudinal axis to thespeed of light (c), or the “velocity factor,” is given by

$\begin{matrix}{{V_{f} = {\frac{v}{c} = \frac{1}{\sqrt{1 + {20\left( \frac{D}{s} \right)^{2.5}\left( \frac{D}{\lambda_{o}} \right)^{0.5}}}}}},} & (45)\end{matrix}$where H is the axial length of the solenoidal helix, D is the coildiameter, N is the number of turns of the coil, s=H/N is theturn-to-turn spacing (or helix pitch) of the coil, and λ_(o) is thefree-space wavelength. Based upon this relationship, the electricallength, or phase delay, of the helical coil is given by

$\begin{matrix}{\theta_{c} = {{\beta_{p}H} = {{\frac{2\pi}{\lambda_{p}}H} = {\frac{2\pi}{V_{f}\lambda_{0}}{H.}}}}} & (46)\end{matrix}$The principle is the same if the helix is wound spirally or is short andfat, but V_(f) and θ_(c) are easier to obtain by experimentalmeasurement. The expression for the characteristic (wave) impedance of ahelical transmission line has also been derived as

$\begin{matrix}{Z_{c} = {{\frac{60}{V_{f}}\left\lbrack {{\ell\;{n\left( \frac{V_{f}\lambda_{0}}{D} \right)}} - 1.027} \right\rbrack}.}} & (47)\end{matrix}$

The spatial phase delay θ_(y) of the structure can be determined usingthe traveling wave phase delay of the vertical feed line conductor 221(FIG. 7). The capacitance of a cylindrical vertical conductor above aprefect ground plane can be expressed as

$\begin{matrix}{{C_{A} = {\frac{2{\pi ɛ}_{o}h_{w}}{{\ell\;{n\left( \frac{h}{a} \right)}} - 1}\mspace{14mu}{Farads}}},} & (48)\end{matrix}$where h_(w) is the vertical length (or height) of the conductor and a isthe radius (in mks units). As with the helical coil, the traveling wavephase delay of the vertical feed line conductor can be given by

$\begin{matrix}{{\theta_{y} = {{\beta_{w}h_{w}} = {{\frac{2\pi}{\lambda_{w}}h_{w}} = {\frac{2\pi}{V_{w}\lambda_{0}}h_{w}}}}},} & (49)\end{matrix}$where β_(w) is the propagation phase constant for the vertical feed lineconductor, h_(w) is the vertical length (or height) of the vertical feedline conductor, V_(w) is the velocity factor on the wire, λ₀ is thewavelength at the supplied frequency, and λ_(w) is the propagationwavelength resulting from the velocity factor V_(w). For a uniformcylindrical conductor, the velocity factor is a constant withV_(w)≈0.94, or in a range from about 0.93 to about 0.98. If the mast isconsidered to be a uniform transmission line, its average characteristicimpedance can be approximated by

$\begin{matrix}{{Z_{w} = {\frac{60}{V_{w}}\left\lbrack {{\ell\;{n\left( \frac{h_{w}}{a} \right)}} - 1} \right\rbrack}},} & (50)\end{matrix}$where V_(w)≈0.94 for a uniform cylindrical conductor and a is the radiusof the conductor. An alternative expression that has been employed inamateur radio literature for the characteristic impedance of asingle-wire feed line can be given by

$\begin{matrix}{Z_{w} = {138{{\log\left( \frac{1.123V_{w}\lambda_{0}}{2\pi\; a} \right)}.}}} & (51)\end{matrix}$Equation (51) implies that Z_(w) for a single-wire feeder varies withfrequency. The phase delay can be determined based upon the capacitanceand characteristic impedance.

With a charge terminal T₁ positioned over the lossy conducting medium203 as shown in FIG. 3, the feed network 209 can be adjusted to excitethe charge terminal T₁ with the phase shift (Φ) of the complex effectiveheight (h_(eff)) equal to the angle (Ψ) of the wave tilt at the Hankelcrossover distance, or Φ=Ψ. When this condition is met, the electricfield produced by the charge oscillating Q₁ on the charge terminal T₁ iscoupled into a guided surface waveguide mode traveling along the surfaceof a lossy conducting medium 203. For example, if the Brewster angle(θ_(i,B)), the phase delay (θ_(y)) associated with the vertical feedline conductor 221 (FIG. 7), and the configuration of the coil 215 (FIG.7) are known, then the position of the tap 224 (FIG. 7) can bedetermined and adjusted to impose an oscillating charge Q₁ on the chargeterminal T₁ with phase Φ=Ψ. The position of the tap 224 may be adjustedto maximize coupling the traveling surface waves into the guided surfacewaveguide mode. Excess coil length beyond the position of the tap 224can be removed to reduce the capacitive effects. The vertical wireheight and/or the geometrical parameters of the helical coil may also bevaried.

The coupling to the guided surface waveguide mode on the surface of thelossy conducting medium 203 can be improved and/or optimized by tuningthe guided surface waveguide probe 200 for standing wave resonance withrespect to a complex image plane associated with the charge Q₁ on thecharge terminal T₁. By doing this, the performance of the guided surfacewaveguide probe 200 can be adjusted for increased and/or maximum voltage(and thus charge Q₁) on the charge terminal T₁. Referring back to FIG.3, the effect of the lossy conducting medium 203 in Region 1 can beexamined using image theory analysis.

Physically, an elevated charge Q₁ placed over a perfectly conductingplane attracts the free charge on the perfectly conducting plane, whichthen “piles up” in the region under the elevated charge Q₁. Theresulting distribution of “bound” electricity on the perfectlyconducting plane is similar to a bell-shaped curve. The superposition ofthe potential of the elevated charge Q₁, plus the potential of theinduced “piled up” charge beneath it, forces a zero equipotentialsurface for the perfectly conducting plane. The boundary value problemsolution that describes the fields in the region above the perfectlyconducting plane may be obtained using the classical notion of imagecharges, where the field from the elevated charge is superimposed withthe field from a corresponding “image” charge below the perfectlyconducting plane.

This analysis may also be used with respect to a lossy conducting medium203 by assuming the presence of an effective image charge Q₁′ beneaththe guided surface waveguide probe 200. The effective image charge Q₁′coincides with the charge Q₁ on the charge terminal T₁ about aconducting image ground plane 130, as illustrated in FIG. 3. However,the image charge Q₁′ is not merely located at some real depth and 180°out of phase with the primary source charge Q₁ on the charge terminalT₁, as they would be in the case of a perfect conductor. Rather, thelossy conducting medium 203 (e.g., a terrestrial medium) presents aphase shifted image. That is to say, the image charge Q₁′ is at acomplex depth below the surface (or physical boundary) of the lossyconducting medium 203. For a discussion of complex image depth,reference is made to Wait, J. R., “Complex Image Theory—Revisited,” IEEEAntennas and Propagation Magazine, Vol. 33, No. 4, August 1991, pp.27-29, which is incorporated herein by reference in its entirety.

Instead of the image charge Q₁′ being at a depth that is equal to thephysical height (H₁) of the charge Q₁, the conducting image ground plane130 (representing a perfect conductor) is located at a complex depth ofz=−d/2 and the image charge Q₁′ appears at a complex depth (i.e., the“depth” has both magnitude and phase), given by −D₁=−(d/2+d/2+H₁)≠H₁.For vertically polarized sources over the Earth,

$\begin{matrix}{{d = {{\frac{2\sqrt{\gamma_{e}^{2} + k_{0}^{2}}}{\gamma_{e}^{2}} \approx \frac{2}{\gamma_{e}}} = {{d_{r} + {j\; d_{i}}} = {{d}{\angle\zeta}}}}},} & (52) \\{where} & \; \\{{\gamma_{e}^{2} = {{j\;{\omega\mu}_{1}\sigma_{1}} - {\omega^{2}\mu_{1}ɛ_{1}}}},{and}} & (53) \\{{k_{o} = {\omega\sqrt{\mu_{o}ɛ_{o}}}},} & (54)\end{matrix}$as indicated in Equation (12). The complex spacing of the image charge,in turn, implies that the external field will experience extra phaseshifts not encountered when the interface is either a dielectric or aperfect conductor. In the lossy conducting medium, the wave front normalis parallel to the tangent of the conducting image ground plane 130 atz=−d/2, and not at the boundary interface between Regions 1 and 2.

Consider the case illustrated in FIG. 8A where the lossy conductingmedium 203 is a finitely conducting Earth 133 with a physical boundary136. The finitely conducting Earth 133 may be replaced by a perfectlyconducting image ground plane 139 as shown in FIG. 8B, which is locatedat a complex depth z₁ below the physical boundary 136. This equivalentrepresentation exhibits the same impedance when looking down into theinterface at the physical boundary 136. The equivalent representation ofFIG. 8B can be modeled as an equivalent transmission line, as shown inFIG. 8C. The cross-section of the equivalent structure is represented asa (z-directed) end-loaded transmission line, with the impedance of theperfectly conducting image plane being a short circuit (z_(s)=0). Thedepth z₁ can be determined by equating the TEM wave impedance lookingdown at the Earth to an image ground plane impedance z_(in) seen lookinginto the transmission line of FIG. 8C.

In the case of FIG. 8A, the propagation constant and wave intrinsicimpedance in the upper region (air) 142 are

$\begin{matrix}{{\gamma_{o} = {{j\;\omega\sqrt{\mu_{o}ɛ_{o}}} = {0 + {j\;\beta_{o}}}}},{and}} & (55) \\{z_{o} = {\frac{j\;{\omega\mu}_{o}}{\gamma_{o}} = {\sqrt{\frac{\mu_{o}}{ɛ_{o}}}.}}} & (56)\end{matrix}$In the lossy Earth 133, the propagation constant and wave intrinsicimpedance are

$\begin{matrix}{{\gamma_{e} = \sqrt{j\;{{\omega\mu}_{1}\left( {\sigma_{1} + {j\;{\omega ɛ}_{1}}} \right)}}},{and}} & (57) \\{Z_{e} = {\frac{j\;{\omega\mu}_{1}}{\gamma_{e}}.}} & (58)\end{matrix}$

For normal incidence, the equivalent representation of FIG. 8B isequivalent to a TEM transmission line whose characteristic impedance isthat of air (z_(o)), with propagation constant of γ_(o), and whoselength is z₁. As such, the image ground plane impedance Z_(in) seen atthe interface for the shorted transmission line of FIG. 8C is given byZ _(in) =Z _(o) tan h(γ_(o) z ₁).  (59)Equating the image ground plane impedance Z_(in) associated with theequivalent model of FIG. 8C to the normal incidence wave impedance ofFIG. 8A and solving for z₁ gives the distance to a short circuit (theperfectly conducting image ground plane 139) as

$\begin{matrix}{{z_{1} = {{\frac{1}{\gamma_{0}}{\tanh^{- 1}\left( \frac{z_{e}}{z_{o}} \right)}} = {{\frac{1}{\gamma_{o}}{\tanh^{- 1}\left( \frac{\gamma_{o}}{\gamma_{e}} \right)}} \approx \frac{1}{\gamma_{e}}}}},} & (60)\end{matrix}$where only the first term of the series expansion for the inversehyperbolic tangent is considered for this approximation. Note that inthe air region 142, the propagation constant is γ_(o)=jβ_(o), soZ_(in)=jZ_(o) tan β_(o)z₁ (which is a purely imaginary quantity for areal z₁), but z_(e) is a complex value if a σ≠0. Therefore, Z_(in)=Z_(e)only when z₁ is a complex distance.

Since the equivalent representation of FIG. 8B includes a perfectlyconducting image ground plane 139, the image depth for a charge orcurrent lying at the surface of the Earth (physical boundary 136) isequal to distance z₁ on the other side of the image ground plane 139, ord=2×z₁ beneath the Earth's surface (which is located at z=0). Thus, thedistance to the perfectly conducting image ground plane 139 can beapproximated by

$\begin{matrix}{d = {{2z_{1}} \approx {\frac{2}{\gamma_{e}}.}}} & (61)\end{matrix}$Additionally, the “image charge” will be “equal and opposite” to thereal charge, so the potential of the perfectly conducting image groundplane 139 at depth z₁=−d/2 will be zero.

If a charge Q₁ is elevated a distance H₁ above the surface of the Earthas illustrated in FIG. 3, then the image charge Q₁′ resides at a complexdistance of D₁=d+H₁ below the surface, or a complex distance of d/2+Hbelow the image ground plane 130. The guided surface waveguide probe 200b of FIG. 7 can be modeled as an equivalent single-wire transmissionline image plane model that can be based upon the perfectly conductingimage ground plane 139 of FIG. 8B. FIG. 9A shows an example of theequivalent single-wire transmission line image plane model, and FIG. 9Billustrates an example of the equivalent classic transmission linemodel, including the shorted transmission line of FIG. 8C.

In the equivalent image plane models of FIGS. 9A and 9B, Φ=θ_(y)+θ_(c)is the traveling wave phase delay of the guided surface waveguide probe200 referenced to Earth 133 (or the lossy conducting medium 203),θ_(c)=β_(p)H is the electrical length of the coil 215 (FIG. 7), ofphysical length H, expressed in degrees, θ_(y)=β_(w)h_(w) is theelectrical length of the vertical feed line conductor 221 (FIG. 7), ofphysical length h_(w), expressed in degrees, and θ_(d)=β_(o) d/2 is thephase shift between the image ground plane 139 and the physical boundary136 of the Earth 133 (or lossy conducting medium 203). In the example ofFIGS. 9A and 9B, Z_(w) is the characteristic impedance of the elevatedvertical feed line conductor 221 in ohms, Z_(c) is the characteristicimpedance of the coil 215 in ohms, and Z_(O) is the characteristicimpedance of free space.

At the base of the guided surface waveguide probe 200, the impedanceseen “looking up” into the structure is Z_(↑)=Z_(base). With a loadimpedance of:

$\begin{matrix}{{Z_{L} = \frac{1}{j\;\omega\; C_{T}}},} & (62)\end{matrix}$where C_(T) is the self-capacitance of the charge terminal T₁, theimpedance seen “looking up” into the vertical feed line conductor 221(FIG. 7) is given by:

$\begin{matrix}{{Z_{2} = {{Z_{W}\frac{Z_{L} + {Z_{W}{\tanh\left( {j\;\beta_{W}h_{W}} \right)}}}{Z_{W} + {Z_{L}{\tanh\left( {j\;\beta_{W}h_{W}} \right)}}}} = {Z_{W}\frac{Z_{L} + {Z_{W}{\tanh\left( {j\;\theta_{y}} \right)}}}{Z_{W} + {Z_{L}{\tanh\left( {j\;\theta_{y}} \right)}}}}}},} & (63)\end{matrix}$and the impedance seen “looking up” into the coil 215 (FIG. 7) is givenby:

$\begin{matrix}{Z_{base} = {{Z_{c}\frac{Z_{2} + {Z_{c}{\tanh\left( {j\;\beta_{p}H} \right)}}}{Z_{c} + {Z_{2}{\tanh\left( {j\;\beta_{p}H} \right)}}}} = {Z_{c}{\frac{Z_{2} + {Z_{c}{\tanh\left( {j\;\theta_{c}} \right)}}}{Z_{c} + {Z_{2}{\tanh\left( {j\;\theta_{c}} \right)}}}.}}}} & (64)\end{matrix}$At the base of the guided surface waveguide probe 200, the impedanceseen “looking down” into the lossy conducting medium 203 isZ_(↓)=Z_(in), which is given by:

$\begin{matrix}{{Z_{in} = {{Z_{o}\frac{Z_{s} + {Z_{o}{\tanh\left\lbrack {j\;{\beta_{o}\left( {d\text{/}2} \right)}} \right\rbrack}}}{Z_{o} + {Z_{s}{\tanh\left\lbrack {j\;{\beta_{o}\left( {d\text{/}2} \right)}} \right\rbrack}}}} = {Z_{o}\mspace{11mu}{\tanh\left( {j\;\theta_{d}} \right)}}}},} & (65)\end{matrix}$where Z_(s)=0.

Neglecting losses, the equivalent image plane model can be tuned toresonance when Z_(↓)+Z_(↑)=0 at the physical boundary 136. Or, in thelow loss case, X_(↓)+X_(↑)=0 at the physical boundary 136, where X isthe corresponding reactive component. Thus, the impedance at thephysical boundary 136 “looking up” into the guided surface waveguideprobe 200 is the conjugate of the impedance at the physical boundary 136“looking down” into the lossy conducting medium 203. By adjusting theload impedance Z_(L) of the charge terminal T₁ while maintaining thetraveling wave phase delay c equal to the angle of the media's wave tiltΨ, so that Φ=Ψ, which improves and/or maximizes coupling of the probe'selectric field to a guided surface waveguide mode along the surface ofthe lossy conducting medium 203 (e.g., Earth), the equivalent imageplane models of FIGS. 9A and 9B can be tuned to resonance with respectto the image ground plane 139. In this way, the impedance of theequivalent complex image plane model is purely resistive, whichmaintains a superposed standing wave on the probe structure thatmaximizes the voltage and elevated charge on terminal T₁, and byequations (1)-(3) and (16) maximizes the propagating surface wave.

It follows from the Hankel solutions, that the guided surface waveexcited by the guided surface waveguide probe 200 is an outwardpropagating traveling wave. The source distribution along the feednetwork 209 between the charge terminal T₁ and the ground stake 218 ofthe guided surface waveguide probe 200 (FIGS. 3 and 7) is actuallycomposed of a superposition of a traveling wave plus a standing wave onthe structure. With the charge terminal T₁ positioned at or above thephysical height h_(p), the phase delay of the traveling wave movingthrough the feed network 209 is matched to the angle of the wave tiltassociated with the lossy conducting medium 203. This mode-matchingallows the traveling wave to be launched along the lossy conductingmedium 203. Once the phase delay has been established for the travelingwave, the load impedance Z_(L) of the charge terminal T₁ is adjusted tobring the probe structure into standing wave resonance with respect tothe image ground plane (130 of FIG. 3 or 139 of FIG. 8), which is at acomplex depth of −d/2. In that case, the impedance seen from the imageground plane has zero reactance and the charge on the charge terminal T₁is maximized.

The distinction between the traveling wave phenomenon and standing wavephenomena is that (1) the phase delay of traveling waves (θ=βd) on asection of transmission line of length d (sometimes called a “delayline”) is due to propagation time delays; whereas (2) theposition-dependent phase of standing waves (which are composed offorward and backward propagating waves) depends on both the line lengthpropagation time delay and impedance transitions at interfaces betweenline sections of different characteristic impedances. In addition to thephase delay that arises due to the physical length of a section oftransmission line operating in sinusoidal steady-state, there is anextra reflection coefficient phase at impedance discontinuities that isdue to the ratio of Z_(oa)/Z_(ob), where Z_(oa) and Z_(ob) are thecharacteristic impedances of two sections of a transmission line suchas, e.g., a helical coil section of characteristic impedanceZ_(oa)=Z_(c) (FIG. 9B) and a straight section of vertical feed lineconductor of characteristic impedance Z_(ob)=Z_(w) (FIG. 9B).

As a result of this phenomenon, two relatively short transmission linesections of widely differing characteristic impedance may be used toprovide a very large phase shift. For example, a probe structurecomposed of two sections of transmission line, one of low impedance andone of high impedance, together totaling a physical length of, say,0.05λ, may be fabricated to provide a phase shift of 90° which isequivalent to a 0.25λ resonance. This is due to the large jump incharacteristic impedances. In this way, a physically short probestructure can be electrically longer than the two physical lengthscombined. This is illustrated in FIGS. 9A and 9B, where thediscontinuities in the impedance ratios provide large jumps in phase.The impedance discontinuity provides a substantial phase shift where thesections are joined together.

Referring to FIG. 10, shown is a flow chart 150 illustrating an exampleof adjusting a guided surface waveguide probe 200 (FIGS. 3 and 7) tosubstantially mode-match to a guided surface waveguide mode on thesurface of the lossy conducting medium, which launches a guided surfacetraveling wave along the surface of a lossy conducting medium 203 (FIG.3). Beginning with 153, the charge terminal T₁ of the guided surfacewaveguide probe 200 is positioned at a defined height above a lossyconducting medium 203. Utilizing the characteristics of the lossyconducting medium 203 and the operating frequency of the guided surfacewaveguide probe 200, the Hankel crossover distance can also be found byequating the magnitudes of Equations (20b) and (21) for −jγρ, andsolving for R_(x) as illustrated by FIG. 4. The complex index ofrefraction (n) can be determined using Equation (41), and the complexBrewster angle (θ_(i,B)) can then be determined from Equation (42). Thephysical height (h_(p)) of the charge terminal T₁ can then be determinedfrom Equation (44). The charge terminal T₁ should be at or higher thanthe physical height (h_(p)) in order to excite the far-out component ofthe Hankel function. This height relationship is initially consideredwhen launching surface waves. To reduce or minimize the bound charge onthe charge terminal T₁, the height should be at least four times thespherical diameter (or equivalent spherical diameter) of the chargeterminal T₁.

At 156, the electrical phase delay (of the elevated charge Q₁ on thecharge terminal T₁ is matched to the complex wave tilt angle Ψ. Thephase delay (θ_(c)) of the helical coil and/or the phase delay (θ_(y))of the vertical feed line conductor can be adjusted to make Φ equal tothe angle (Ψ) of the wave tilt (W). Based on Equation (31), the angle(Ψ) of the wave tilt can be determined from:

$\begin{matrix}{W = {\frac{E_{\rho}}{E_{z}} = {\frac{1}{\tan\;\theta_{i,B}} = {\frac{1}{n} = {{W}{e^{j\;\Psi}.}}}}}} & (66)\end{matrix}$The electrical phase Φ can then be matched to the angle of the wavetilt. This angular (or phase) relationship is next considered whenlaunching surface waves. For example, the electrical phase delayΦ=θ_(c)+θ_(y) can be adjusted by varying the geometrical parameters ofthe coil 215 (FIG. 7) and/or the length (or height) of the vertical feedline conductor 221 (FIG. 7). By matching Φ=Ψ, an electric field can beestablished at or beyond the Hankel crossover distance (R_(x)) with acomplex Brewster angle at the boundary interface to excite the surfacewaveguide mode and launch a traveling wave along the lossy conductingmedium 203.

Next at 159, the load impedance of the charge terminal T₁ is tuned toresonate the equivalent image plane model of the guided surfacewaveguide probe 200. The depth (d/2) of the conducting image groundplane 139 of FIGS. 9A and 9B (or 130 of FIG. 3) can be determined usingEquations (52), (53) and (54) and the values of the lossy conductingmedium 203 (e.g., the Earth), which can be measured. Using that depth,the phase shift (θ_(d)) between the image ground plane 139 and thephysical boundary 136 of the lossy conducting medium 203 can bedetermined using θ_(d)=β_(o) d/2. The impedance (Z_(in)) as seen“looking down” into the lossy conducting medium 203 can then bedetermined using Equation (65). This resonance relationship can beconsidered to maximize the launched surface waves.

Based upon the adjusted parameters of the coil 215 and the length of thevertical feed line conductor 221, the velocity factor, phase delay, andimpedance of the coil 215 and vertical feed line conductor 221 can bedetermined using Equations (45) through (51). In addition, theself-capacitance (C_(T)) of the charge terminal T₁ can be determinedusing, e.g., Equation (24). The propagation factor (β_(p)) of the coil215 can be determined using Equation (35) and the propagation phaseconstant (β_(w)) for the vertical feed line conductor 221 can bedetermined using Equation (49). Using the self-capacitance and thedetermined values of the coil 215 and vertical feed line conductor 221,the impedance (Z_(base)) of the guided surface waveguide probe 200 asseen “looking up” into the coil 215 can be determined using Equations(62), (63) and (64).

The equivalent image plane model of the guided surface waveguide probe200 can be tuned to resonance by adjusting the load impedance Z_(L) suchthat the reactance component X_(base) of Z_(base) cancels out thereactance component X_(in) of Z_(in), or X_(base)+X_(in)=0. Thus, theimpedance at the physical boundary 136 “looking up” into the guidedsurface waveguide probe 200 is the conjugate of the impedance at thephysical boundary 136 “looking down” into the lossy conducting medium203. The load impedance Z_(L) can be adjusted by varying the capacitance(C_(T)) of the charge terminal T₁ without changing the electrical phasedelay Φ=θ_(c)+θ_(y) of the charge terminal T₁. An iterative approach maybe taken to tune the load impedance Z_(L) for resonance of theequivalent image plane model with respect to the conducting image groundplane 139 (or 130). In this way, the coupling of the electric field to aguided surface waveguide mode along the surface of the lossy conductingmedium 203 (e.g., Earth) can be improved and/or maximized.

This may be better understood by illustrating the situation with anumerical example. Consider a guided surface waveguide probe 200comprising a top-loaded vertical stub of physical height h_(p) with acharge terminal T₁ at the top, where the charge terminal T₁ is excitedthrough a helical coil and vertical feed line conductor at anoperational frequency (f_(o)) of 1.85 MHz. With a height (H₁) of 16 feetand the lossy conducting medium 203 (e.g., Earth) having a relativepermittivity of ε_(r)=15 and a conductivity of σ₁=0.010 mhos/m, severalsurface wave propagation parameters can be calculated for f_(o)=1.850MHz. Under these conditions, the Hankel crossover distance can be foundto be R_(x)=54.5 feet with a physical height of h_(p)=5.5 feet, which iswell below the actual height of the charge terminal T₁. While a chargeterminal height of H₁=5.5 feet could have been used, the taller probestructure reduced the bound capacitance, permitting a greater percentageof free charge on the charge terminal T₁ providing greater fieldstrength and excitation of the traveling wave.

The wave length can be determined as:

$\begin{matrix}{{\lambda_{O} = {\frac{c}{f_{o}} = {162.162\mspace{20mu}{meters}}}},} & (67)\end{matrix}$where c is the speed of light. The complex index of refraction is:n=√{square root over (ε_(r) −jx)}=7.529−j6.546,  (68)from Equation (41), where x=σ₁/ωε_(o) with ω=2πf_(o), and the complexBrewster angle is:θ_(i,B)=arctan(√{square root over (ε_(r) −jx)})=85.6−j3.744°.  (69)from Equation (42). Using Equation (66), the wave tilt values can bedetermined to be:

$\begin{matrix}{W = {\frac{1}{\tan\;\theta_{i,B}} = {\frac{1}{n} = {{{W}e^{j\;\Psi}} = {0.101\mspace{14mu}{e^{j\; 40.614{^\circ}}.}}}}}} & (70)\end{matrix}$Thus, the helical coil can be adjusted to match Φ=Ψ=40.614°

The velocity factor of the vertical feed line conductor (approximated asa uniform cylindrical conductor with a diameter of 0.27 inches) can begiven as V_(w)≈0.93. Since h_(p)<<λ_(o), the propagation phase constantfor the vertical feed line conductor can be approximated as:

$\begin{matrix}{\beta_{w} = {\frac{2\pi}{\lambda_{w}} = {\frac{2\pi}{V_{w}\lambda_{0}} = {0.042\mspace{14mu}{m^{- 1}.}}}}} & (71)\end{matrix}$From Equation (49) the phase delay of the vertical feed line conductoris:θ_(y)=β_(w) h _(w)≈β_(w) h _(p)=11.640°.  (72)By adjusting the phase delay of the helical coil so thatθ_(c)=28.974°=40.614°−11.640°, Φ will equal Ψ to match the guidedsurface waveguide mode. To illustrate the relationship between Φ and Ψ,FIG. 11 shows a plot of both over a range of frequencies. As both Φ andΨ are frequency dependent, it can be seen that their respective curvescross over each other at approximately 1.85 MHz.

For a helical coil having a conductor diameter of 0.0881 inches, a coildiameter (D) of 30 inches and a turn-to-turn spacing (s) of 4 inches,the velocity factor for the coil can be determined using Equation (45)as:

$\begin{matrix}{{V_{f} = {\frac{1}{\sqrt{1 + {20\left( \frac{D}{s} \right)^{2.5}\left( \frac{D}{\lambda_{o}} \right)^{0.5}}}} = 0.069}},} & (73)\end{matrix}$and the propagation factor from Equation (35) is:

$\begin{matrix}{\beta_{p} = {\frac{2\pi}{V_{f}\lambda_{0}} = {0.564\mspace{20mu}{m^{- 1}.}}}} & (74)\end{matrix}$

With θ_(c)=28.974°, the axial length of the solenoidal helix (H) can bedetermined using Equation (46) such that:

$\begin{matrix}{H = {\frac{\theta_{c}}{\beta_{p}} = {35.2732\mspace{14mu}{{inches}.}}}} & (75)\end{matrix}$This height determines the location on the helical coil where thevertical feed line conductor is connected, resulting in a coil with8.818 turns (N=H/s).

With the traveling wave phase delay of the coil and vertical feed lineconductor adjusted to match the wave tilt angle (Φ=θ_(c)+θ_(y)=Ψ), theload impedance (Z_(L)) of the charge terminal T₁ can be adjusted forstanding wave resonance of the equivalent image plane model of theguided surface wave probe 200. From the measured permittivity,conductivity and permeability of the Earth, the radial propagationconstant can be determined using Equation (57)γ_(e)=√{square root over (jωμu ₁(σ₁ +jωε ₁))}=0.25+j0.292 m⁻¹,  (76)And the complex depth of the conducting image ground plane can beapproximated from Equation (52) as:

$\begin{matrix}{{{d \approx \frac{2}{\gamma_{e}}} = {3.364 + {j\mspace{11mu} 3.963\mspace{14mu}{meters}}}},} & (77)\end{matrix}$with a corresponding phase shift between the conducting image groundplane and the physical boundary of the Earth given by:θ_(d)=β_(o)(d/2)=4.015−j4.73°.  (78)Using Equation (65), the impedance seen “looking down” into the lossyconducting medium 203 (i.e., Earth) can be determined as:Z _(in) =Z _(o) tan h(jθ _(d))=R _(in) +jX _(n)=31.191+j26.27ohms.  (79)

By matching the reactive component (X_(in)) seen “looking down” into thelossy conducting medium 203 with the reactive component (X_(base)) seen“looking up” into the guided surface wave probe 200, the coupling intothe guided surface waveguide mode may be maximized. This can beaccomplished by adjusting the capacitance of the charge terminal T₁without changing the traveling wave phase delays of the coil andvertical feed line conductor. For example, by adjusting the chargeterminal capacitance (C_(T)) to 61.8126 pF, the load impedance fromEquation (62) is:

$\begin{matrix}{{Z_{L} = {\frac{1}{j\;\omega\; C_{T}} = {{- j}\mspace{14mu} 1392\mspace{14mu}{ohms}}}},} & (80)\end{matrix}$and the reactive components at the boundary are matched.

Using Equation (51), the impedance of the vertical feed line conductor(having a diameter (2a) of 0.27 inches) is given as

$\begin{matrix}{{Z_{w} = {{138\mspace{14mu}{\log\left( \frac{1.123\; V_{w}\lambda_{0}}{2\pi\; a} \right)}} = {537.534\mspace{14mu}{ohms}}}},} & (81)\end{matrix}$and the impedance seen “looking up” into the vertical feed lineconductor is given by Equation (63) as:

$\begin{matrix}{Z_{2} = {{Z_{W}\frac{Z_{L} + {Z_{W}{\tanh\left( {j\;\theta_{y}} \right)}}}{Z_{W} + {Z_{L}{\tanh\left( {j\;\theta_{y}} \right)}}}} = {{- j}\mspace{11mu} 835.438\mspace{14mu}{{ohms}.}}}} & (82)\end{matrix}$Using Equation (47), the characteristic impedance of the helical coil isgiven as

$\begin{matrix}{{Z_{c} = {{\frac{60}{V_{f}}\left\lbrack {{\ell\;{n\left( \frac{V_{f}\lambda_{0}}{D} \right)}} - 1.027} \right\rbrack} = {1446\mspace{14mu}{ohms}}}},} & (83)\end{matrix}$and the impedance seen “looking up” into the coil at the base is givenby Equation (64) as:

$\begin{matrix}{Z_{base} = {{Z_{c}\frac{Z_{2} + {Z_{c}{\tanh\left( {j\;\theta_{c}} \right)}}}{Z_{c} + {Z_{2}{\tanh\left( {j\;\theta_{c}} \right)}}}} = {{- j}\mspace{11mu} 26.271\mspace{14mu}{{ohms}.}}}} & (84)\end{matrix}$When compared to the solution of Equation (79), it can be seen that thereactive components are opposite and approximately equal, and thus areconjugates of each other. Thus, the impedance (Z_(ip)) seen “looking up”into the equivalent image plane model of FIGS. 9A and 9B from theperfectly conducting image ground plane is only resistive orZ_(ip)=R+j0.

When the electric fields produced by a guided surface waveguide probe200 (FIG. 3) are established by matching the traveling wave phase delayof the feed network to the wave tilt angle and the probe structure isresonated with respect to the perfectly conducting image ground plane atcomplex depth z=−d/2, the fields are substantially mode-matched to aguided surface waveguide mode on the surface of the lossy conductingmedium, a guided surface traveling wave is launched along the surface ofthe lossy conducting medium. As illustrated in FIG. 1, the guided fieldstrength curve 103 of the guided electromagnetic field has acharacteristic exponential decay of e^(−αd)/√{square root over (d)} andexhibits a distinctive knee 109 on the log-log scale.

In summary, both analytically and experimentally, the traveling wavecomponent on the structure of the guided surface waveguide probe 200 hasa phase delay (Φ) at its upper terminal that matches the angle (Ψ) ofthe wave tilt of the surface traveling wave (Φ=Ψ). Under this condition,the surface waveguide may be considered to be “mode-matched”.Furthermore, the resonant standing wave component on the structure ofthe guided surface waveguide probe 200 has a V_(MAX) at the chargeterminal T₁ and a V_(MIN) down at the image plane 139 (FIG. 8B) whereZ_(ip)=R_(ip)+j0 at a complex depth of z=−d/2, not at the connection atthe physical boundary 136 of the lossy conducting medium 203 (FIG. 8B).Lastly, the charge terminal T₁ is of sufficient height H₁ of FIG. 3(h≥R_(x) tan ψ_(i,B)) so that electromagnetic waves incident onto thelossy conducting medium 203 at the complex Brewster angle do so out at adistance (≥R_(x)) where the 1/√{square root over (r)} term ispredominant. Receive circuits can be utilized with one or more guidedsurface waveguide probes to facilitate wireless transmission and/orpower delivery systems.

Referring back to FIG. 3, operation of a guided surface waveguide probe200 may be controlled to adjust for variations in operational conditionsassociated with the guided surface waveguide probe 200. For example, anadaptive probe control system 230 can be used to control the feednetwork 209 and/or the charge terminal T₁ to control the operation ofthe guided surface waveguide probe 200. Operational conditions caninclude, but are not limited to, variations in the characteristics ofthe lossy conducting medium 203 (e.g., conductivity σ and relativepermittivity ε_(r)), variations in field strength and/or variations inloading of the guided surface waveguide probe 200. As can be seen fromEquations (31), (41) and (42), the index of refraction (n), the complexBrewster angle (θ_(i,B)), and the wave tilt (|w|e^(jΨ)) can be affectedby changes in soil conductivity and permittivity resulting from, e.g.,weather conditions.

Equipment such as, e.g., conductivity measurement probes, permittivitysensors, ground parameter meters, field meters, current monitors and/orload receivers can be used to monitor for changes in the operationalconditions and provide information about current operational conditionsto the adaptive probe control system 230. The probe control system 230can then make one or more adjustments to the guided surface waveguideprobe 200 to maintain specified operational conditions for the guidedsurface waveguide probe 200. For instance, as the moisture andtemperature vary, the conductivity of the soil will also vary.Conductivity measurement probes and/or permittivity sensors may belocated at multiple locations around the guided surface waveguide probe200. Generally, it would be desirable to monitor the conductivity and/orpermittivity at or about the Hankel crossover distance R_(x) for theoperational frequency. Conductivity measurement probes and/orpermittivity sensors may be located at multiple locations (e.g., in eachquadrant) around the guided surface waveguide probe 200.

The conductivity measurement probes and/or permittivity sensors can beconfigured to evaluate the conductivity and/or permittivity on aperiodic basis and communicate the information to the probe controlsystem 230. The information may be communicated to the probe controlsystem 230 through a network such as, but not limited to, a LAN, WLAN,cellular network, or other appropriate wired or wireless communicationnetwork. Based upon the monitored conductivity and/or permittivity, theprobe control system 230 may evaluate the variation in the index ofrefraction (n), the complex Brewster angle (θ_(i,B)), and/or the wavetilt (|W|e^(jΨ)) and adjust the guided surface waveguide probe 200 tomaintain the phase delay (Φ) of the feed network 209 equal to the wavetilt angle (Ψ) and/or maintain resonance of the equivalent image planemodel of the guided surface waveguide probe 200. This can beaccomplished by adjusting, e.g., θ_(y), θ_(c) and/or C_(T). Forinstance, the probe control system 230 can adjust the self-capacitanceof the charge terminal T₁ and/or the phase delay (θ_(y), θ_(c)) appliedto the charge terminal T₁ to maintain the electrical launchingefficiency of the guided surface wave at or near its maximum. Forexample, the self-capacitance of the charge terminal T₁ can be varied bychanging the size of the terminal. The charge distribution can also beimproved by increasing the size of the charge terminal T₁, which canreduce the chance of an electrical discharge from the charge terminalT₁. In other embodiments, the charge terminal T₁ can include a variableinductance that can be adjusted to change the load impedance Z_(L). Thephase applied to the charge terminal T₁ can be adjusted by varying thetap position on the coil 215 (FIG. 7), and/or by including a pluralityof predefined taps along the coil 215 and switching between thedifferent predefined tap locations to maximize the launching efficiency.

Field or field strength (FS) meters may also be distributed about theguided surface waveguide probe 200 to measure field strength of fieldsassociated with the guided surface wave. The field or FS meters can beconfigured to detect the field strength and/or changes in the fieldstrength (e.g., electric field strength) and communicate thatinformation to the probe control system 230. The information may becommunicated to the probe control system 230 through a network such as,but not limited to, a LAN, WLAN, cellular network, or other appropriatecommunication network. As the load and/or environmental conditionschange or vary during operation, the guided surface waveguide probe 200may be adjusted to maintain specified field strength(s) at the FS meterlocations to ensure appropriate power transmission to the receivers andthe loads they supply.

For example, the phase delay (Φ=θ_(y)+θ_(c)) applied to the chargeterminal T₁ can be adjusted to match the wave tilt angle (Ψ). Byadjusting one or both phase delays, the guided surface waveguide probe200 can be adjusted to ensure the wave tilt corresponds to the complexBrewster angle. This can be accomplished by adjusting a tap position onthe coil 215 (FIG. 7) to change the phase delay supplied to the chargeterminal T₁. The voltage level supplied to the charge terminal T₁ canalso be increased or decreased to adjust the electric field strength.This may be accomplished by adjusting the output voltage of theexcitation source 212 or by adjusting or reconfiguring the feed network209. For instance, the position of the tap 227 (FIG. 7) for the ACsource 212 can be adjusted to increase the voltage seen by the chargeterminal T₁. Maintaining field strength levels within predefined rangescan improve coupling by the receivers, reduce ground current losses, andavoid interference with transmissions from other guided surfacewaveguide probes 200.

The probe control system 230 can be implemented with hardware, firmware,software executed by hardware, or a combination thereof. For example,the probe control system 230 can include processing circuitry includinga processor and a memory, both of which can be coupled to a localinterface such as, for example, a data bus with an accompanyingcontrol/address bus as can be appreciated by those with ordinary skillin the art. A probe control application may be executed by the processorto adjust the operation of the guided surface waveguide probe 200 basedupon monitored conditions. The probe control system 230 can also includeone or more network interfaces for communicating with the variousmonitoring devices. Communications can be through a network such as, butnot limited to, a LAN, WLAN, cellular network, or other appropriatecommunication network. The probe control system 230 may comprise, forexample, a computer system such as a server, desktop computer, laptop,or other system with like capability.

Referring back to the example of FIG. 5A, the complex angle trigonometryis shown for the ray optic interpretation of the incident electric field(E) of the charge terminal T₁ with a complex Brewster angle (θ_(i,B)) atthe Hankel crossover distance (R_(x)). Recall that, for a lossyconducting medium, the Brewster angle is complex and specified byequation (38). Electrically, the geometric parameters are related by theelectrical effective height (h_(eff)) of the charge terminal T₁ byequation (39). Since both the physical height (h_(p)) and the Hankelcrossover distance (R_(x)) are real quantities, the angle of the desiredguided surface wave tilt at the Hankel crossover distance (W_(Rx)) isequal to the phase (Φ) of the complex effective height (h_(eff)). Withthe charge terminal T₁ positioned at the physical height h_(p) andexcited with a charge having the appropriate phase Φ, the resultingelectric field is incident with the lossy conducting medium boundaryinterface at the Hankel crossover distance R_(x), and at the Brewsterangle. Under these conditions, the guided surface waveguide mode can beexcited without reflection or substantially negligible reflection.

However, Equation (39) means that the physical height of the guidedsurface waveguide probe 200 can be relatively small. While this willexcite the guided surface waveguide mode, this can result in an undulylarge bound charge with little free charge. To compensate, the chargeterminal T₁ can be raised to an appropriate elevation to increase theamount of free charge. As one example rule of thumb, the charge terminalT₁ can be positioned at an elevation of about 4-5 times (or more) theeffective diameter of the charge terminal T₁. FIG. 6 illustrates theeffect of raising the charge terminal T₁ above the physical height(h_(p)) shown in FIG. 5A. The increased elevation causes the distance atwhich the wave tilt is incident with the lossy conductive medium to movebeyond the Hankel crossover point 121 (FIG. 5A). To improve coupling inthe guided surface waveguide mode, and thus provide for a greaterlaunching efficiency of the guided surface wave, a lower compensationterminal T₂ can be used to adjust the total effective height (h_(TE)) ofthe charge terminal T₁ such that the wave tilt at the Hankel crossoverdistance is at the Brewster angle.

Referring to FIG. 12, shown is an example of a guided surface waveguideprobe 200 c that includes an elevated charge terminal T₁ and a lowercompensation terminal T₂ that are arranged along a vertical axis z thatis normal to a plane presented by the lossy conducting medium 203. Inthis respect, the charge terminal T₁ is placed directly above thecompensation terminal T₂ although it is possible that some otherarrangement of two or more charge and/or compensation terminals T_(N)can be used. The guided surface waveguide probe 200 c is disposed abovea lossy conducting medium 203 according to an embodiment of the presentdisclosure. The lossy conducting medium 203 makes up Region 1 with asecond medium 206 that makes up Region 2 sharing a boundary interfacewith the lossy conducting medium 203.

The guided surface waveguide probe 200 c includes a feed network 209that couples an excitation source 212 to the charge terminal T₁ and thecompensation terminal T₂. According to various embodiments, charges Q₁and Q₂ can be imposed on the respective charge and compensationterminals T₁ and T₂, depending on the voltages applied to terminals T₁and T₂ at any given instant. I₁ is the conduction current feeding thecharge Q₁ on the charge terminal T₁ via the terminal lead, and I₂ is theconduction current feeding the charge Q₂ on the compensation terminal T₂via the terminal lead.

According to the embodiment of FIG. 12, the charge terminal T₁ ispositioned over the lossy conducting medium 203 at a physical height H₁,and the compensation terminal T₂ is positioned directly below T₁ alongthe vertical axis z at a physical height H₂, where H₂ is less than H₁.The height h of the transmission structure may be calculated as h=H₁−H₂.The charge terminal T₁ has an isolated (or self) capacitance C₁, and thecompensation terminal T₂ has an isolated (or self) capacitance C₂. Amutual capacitance C_(M) can also exist between the terminals T₁ and T₂depending on the distance therebetween. During operation, charges Q₁ andQ₂ are imposed on the charge terminal T₁ and the compensation terminalT₂, respectively, depending on the voltages applied to the chargeterminal T₁ and the compensation terminal T₂ at any given instant.

Referring next to FIG. 13, shown is a ray optics interpretation of theeffects produced by the elevated charge Q₁ on charge terminal T₁ andcompensation terminal T₂ of FIG. 12. With the charge terminal T₁elevated to a height where the ray intersects with the lossy conductivemedium at the Brewster angle at a distance greater than the Hankelcrossover point 121 as illustrated by line 163, the compensationterminal T₂ can be used to adjust h_(TE) by compensating for theincreased height. The effect of the compensation terminal T₂ is toreduce the electrical effective height of the guided surface waveguideprobe (or effectively raise the lossy medium interface) such that thewave tilt at the Hankel crossover distance is at the Brewster angle asillustrated by line 166.

The total effective height can be written as the superposition of anupper effective height (h_(UE)) associated with the charge terminal T₁and a lower effective height (h_(LE)) associated with the compensationterminal T₂ such thath _(TE) =h _(UE) +h _(LE) =h _(p) e ^(j(βh) ^(p) ^(+Φ) ^(U) ^()+h) _(d)e ^(j(βh) ^(d) ^(+Φ) ^(L) ⁾ =R _(x) ×W,  (85)where Φ_(U) is the phase delay applied to the upper charge terminal T₁,Φ_(L) is the phase delay applied to the lower compensation terminal T₂,β=2π/λ_(p) is the propagation factor from Equation (35), h_(p) is thephysical height of the charge terminal T₁ and h_(d) is the physicalheight of the compensation terminal T₂. If extra lead lengths are takeninto consideration, they can be accounted for by adding the chargeterminal lead length z to the physical height h_(p) of the chargeterminal T₁ and the compensation terminal lead length y to the physicalheight h_(d) of the compensation terminal T₂ as shown inh _(TE)=(h _(p) +z)e ^(j(β(h) ^(p) ^(+z)+Φ) ^(U) ⁾+(h _(d) +y)e ^(j(β(h)^(d) ^(+Y)+Φ) ^(L) ⁾ =R _(x) ×W.  (86)The lower effective height can be used to adjust the total effectiveheight (h_(TE)) to equal the complex effective height (h_(eff)) of FIG.5A.

Equations (85) or (86) can be used to determine the physical height ofthe lower disk of the compensation terminal T₂ and the phase angles tofeed the terminals in order to obtain the desired wave tilt at theHankel crossover distance. For example, Equation (86) can be rewrittenas the phase shift applied to the charge terminal T₁ as a function ofthe compensation terminal height (h_(d)) to give

$\begin{matrix}{{\Phi_{U}\left( h_{d} \right)} = {{- {\beta\left( {h_{p} + z} \right)}} - {j\;{{\ln\left( \frac{{R_{x} \times W} - {\left( {h_{d} + y} \right)e^{j{({{\beta\; h_{d}} + {\beta\; y} + \Phi_{L}})}}}}{\left( {h_{p} + z} \right)} \right)}.}}}} & (87)\end{matrix}$

To determine the positioning of the compensation terminal T₂, therelationships discussed above can be utilized. First, the totaleffective height (h_(TE)) is the superposition of the complex effectiveheight (h_(UE)) of the upper charge terminal T₁ and the complexeffective height (h_(LE)) of the lower compensation terminal T₂ asexpressed in Equation (86). Next, the tangent of the angle of incidencecan be expressed geometrically as

$\begin{matrix}{{{\tan\;\psi_{E}} = \frac{h_{TE}}{R_{x}}},} & (88)\end{matrix}$which is equal to the definition of the wave tilt, W. Finally, given thedesired Hankel crossover distance R_(x), the h_(TE) can be adjusted tomake the wave tilt of the incident ray match the complex Brewster angleat the Hankel crossover point 121. This can be accomplished by adjustingh_(p), Φ_(U), and/or h_(d).

These concepts may be better understood when discussed in the context ofan example of a guided surface waveguide probe. Referring to FIG. 14,shown is a graphical representation of an example of a guided surfacewaveguide probe 200 d including an upper charge terminal T₁ (e.g., asphere at height h_(T)) and a lower compensation terminal T₂ (e.g., adisk at height h_(d)) that are positioned along a vertical axis z thatis substantially normal to the plane presented by the lossy conductingmedium 203. During operation, charges Q₁ and Q₂ are imposed on thecharge and compensation terminals T₁ and T₂, respectively, depending onthe voltages applied to the terminals T₁ and T₂ at any given instant.

An AC source 212 acts as the excitation source for the charge terminalT₁, which is coupled to the guided surface waveguide probe 200 d througha feed network 209 comprising a coil 215 such as, e.g., a helical coil.The AC source 212 can be connected across a lower portion of the coil215 through a tap 227, as shown in FIG. 14, or can be inductivelycoupled to the coil 215 by way of a primary coil. The coil 215 can becoupled to a ground stake 218 at a first end and the charge terminal T₁at a second end. In some implementations, the connection to the chargeterminal T₁ can be adjusted using a tap 224 at the second end of thecoil 215. The compensation terminal T₂ is positioned above andsubstantially parallel with the lossy conducting medium 203 (e.g., theground or Earth), and energized through a tap 233 coupled to the coil215. An ammeter 236 located between the coil 215 and ground stake 218can be used to provide an indication of the magnitude of the currentflow (I₀) at the base of the guided surface waveguide probe.Alternatively, a current clamp may be used around the conductor coupledto the ground stake 218 to obtain an indication of the magnitude of thecurrent flow (I₀).

In the example of FIG. 14, the coil 215 is coupled to a ground stake 218at a first end and the charge terminal T₁ at a second end via a verticalfeed line conductor 221. In some implementations, the connection to thecharge terminal T₁ can be adjusted using a tap 224 at the second end ofthe coil 215 as shown in FIG. 14. The coil 215 can be energized at anoperating frequency by the AC source 212 through a tap 227 at a lowerportion of the coil 215. In other implementations, the AC source 212 canbe inductively coupled to the coil 215 through a primary coil. Thecompensation terminal T₂ is energized through a tap 233 coupled to thecoil 215. An ammeter 236 located between the coil 215 and ground stake218 can be used to provide an indication of the magnitude of the currentflow at the base of the guided surface waveguide probe 200 d.Alternatively, a current clamp may be used around the conductor coupledto the ground stake 218 to obtain an indication of the magnitude of thecurrent flow. The compensation terminal T₂ is positioned above andsubstantially parallel with the lossy conducting medium 203 (e.g., theground).

In the example of FIG. 14, the connection to the charge terminal T₁located on the coil 215 above the connection point of tap 233 for thecompensation terminal T₂. Such an adjustment allows an increased voltage(and thus a higher charge Q₁) to be applied to the upper charge terminalT₁. In other embodiments, the connection points for the charge terminalT₁ and the compensation terminal T₂ can be reversed. It is possible toadjust the total effective height (h_(TE)) of the guided surfacewaveguide probe 200 d to excite an electric field having a guidedsurface wave tilt at the Hankel crossover distance R_(x). The Hankelcrossover distance can also be found by equating the magnitudes ofequations (20b) and (21) for −jγρ, and solving for R_(x) as illustratedby FIG. 4. The index of refraction (n), the complex Brewster angle(θ_(i,B) and ψ_(i,B)), the wave tilt (|W|e^(jΨ)) and the complexeffective height (h_(eff)=h_(p)e^(jΦ)) can be determined as describedwith respect to Equations (41)-(44) above.

With the selected charge terminal T₁ configuration, a spherical diameter(or the effective spherical diameter) can be determined. For example, ifthe charge terminal T₁ is not configured as a sphere, then the terminalconfiguration may be modeled as a spherical capacitance having aneffective spherical diameter. The size of the charge terminal T₁ can bechosen to provide a sufficiently large surface for the charge Q₁ imposedon the terminals. In general, it is desirable to make the chargeterminal T₁ as large as practical. The size of the charge terminal T₁should be large enough to avoid ionization of the surrounding air, whichcan result in electrical discharge or sparking around the chargeterminal. To reduce the amount of bound charge on the charge terminalT₁, the desired elevation to provide free charge on the charge terminalT₁ for launching a guided surface wave should be at least 4-5 times theeffective spherical diameter above the lossy conductive medium (e.g.,the Earth). The compensation terminal T₂ can be used to adjust the totaleffective height (h_(TE)) of the guided surface waveguide probe 200 d toexcite an electric field having a guided surface wave tilt at R_(x). Thecompensation terminal T₂ can be positioned below the charge terminal T₁at h_(d)=h_(T)−h_(p), where h_(T) is the total physical height of thecharge terminal T₁. With the position of the compensation terminal T₂fixed and the phase delay Φ_(U) applied to the upper charge terminal T₁,the phase delay Φ_(L) applied to the lower compensation terminal T₂ canbe determined using the relationships of Equation (86), such that:

$\begin{matrix}{{\Phi_{U}\left( h_{d} \right)} = {{- {\beta\left( {h_{d} + y} \right)}} - {j\;{{\ln\left( \frac{{R_{x} \times W} - {\left( {h_{p} + z} \right)e^{j{({{\beta\; h_{p}} + {\beta\; z} + \Phi_{L}})}}}}{\left( {h_{d} + y} \right)} \right)}.}}}} & (89)\end{matrix}$In alternative embodiments, the compensation terminal T₂ can bepositioned at a height h_(d) where Im{Φ_(L)}=0. This is graphicallyillustrated in FIG. 15A, which shows plots 172 and 175 of the imaginaryand real parts of Φ_(U), respectively. The compensation terminal T₂ ispositioned at a height h_(d) where Im{Φ_(U)}=0, as graphicallyillustrated in plot 172. At this fixed height, the coil phase Φ_(U) canbe determined from Re{Φ_(U)}, as graphically illustrated in plot 175.

With the AC source 212 coupled to the coil 215 (e.g., at the 50Ω pointto maximize coupling), the position of tap 233 may be adjusted forparallel resonance of the compensation terminal T₂ with at least aportion of the coil at the frequency of operation. FIG. 15B shows aschematic diagram of the general electrical hookup of FIG. 14 in whichV₁ is the voltage applied to the lower portion of the coil 215 from theAC source 212 through tap 227, V₂ is the voltage at tap 224 that issupplied to the upper charge terminal T₁, and V₃ is the voltage appliedto the lower compensation terminal T₂ through tap 233. The resistancesR_(p) and R_(d) represent the ground return resistances of the chargeterminal T₁ and compensation terminal T₂, respectively. The charge andcompensation terminals T₁ and T₂ may be configured as spheres,cylinders, toroids, rings, hoods, or any other combination of capacitivestructures. The size of the charge and compensation terminals T₁ and T₂can be chosen to provide a sufficiently large surface for the charges Q₁and Q₂ imposed on the terminals. In general, it is desirable to make thecharge terminal T₁ as large as practical. The size of the chargeterminal T₁ should be large enough to avoid ionization of thesurrounding air, which can result in electrical discharge or sparkingaround the charge terminal. The self-capacitance C_(p) and C_(d) of thecharge and compensation terminals T₁ and T₂ respectively, can bedetermined using, for example, equation (24).

As can be seen in FIG. 15B, a resonant circuit is formed by at least aportion of the inductance of the coil 215, the self-capacitance C_(d) ofthe compensation terminal T₂, and the ground return resistance R_(d)associated with the compensation terminal T₂. The parallel resonance canbe established by adjusting the voltage V₃ applied to the compensationterminal T₂ (e.g., by adjusting a tap 233 position on the coil 215) orby adjusting the height and/or size of the compensation terminal T₂ toadjust C_(d). The position of the coil tap 233 can be adjusted forparallel resonance, which will result in the ground current through theground stake 218 and through the ammeter 236 reaching a maximum point.After parallel resonance of the compensation terminal T₂ has beenestablished, the position of the tap 227 for the AC source 212 can beadjusted to the 500 point on the coil 215.

Voltage V₂ from the coil 215 can be applied to the charge terminal T₁,and the position of tap 224 can be adjusted such that the phase (Φ) ofthe total effective height (h_(TE)) approximately equals the angle ofthe guided surface wave tilt (W_(Rx)) at the Hankel crossover distance(R_(x)). The position of the coil tap 224 can be adjusted until thisoperating point is reached, which results in the ground current throughthe ammeter 236 increasing to a maximum. At this point, the resultantfields excited by the guided surface waveguide probe 200 d aresubstantially mode-matched to a guided surface waveguide mode on thesurface of the lossy conducting medium 203, resulting in the launchingof a guided surface wave along the surface of the lossy conductingmedium 203. This can be verified by measuring field strength along aradial extending from the guided surface waveguide probe 200.

Resonance of the circuit including the compensation terminal T₂ maychange with the attachment of the charge terminal T₁ and/or withadjustment of the voltage applied to the charge terminal T₁ through tap224. While adjusting the compensation terminal circuit for resonanceaids the subsequent adjustment of the charge terminal connection, it isnot necessary to establish the guided surface wave tilt (W_(Rx)) at theHankel crossover distance (R_(x)). The system may be further adjusted toimprove coupling by iteratively adjusting the position of the tap 227for the AC source 212 to be at the 50Ω point on the coil 215 andadjusting the position of tap 233 to maximize the ground current throughthe ammeter 236. Resonance of the circuit including the compensationterminal T₂ may drift as the positions of taps 227 and 233 are adjusted,or when other components are attached to the coil 215.

In other implementations, the voltage V₂ from the coil 215 can beapplied to the charge terminal T₁, and the position of tap 233 can beadjusted such that the phase (Φ) of the total effective height (h_(TE))approximately equals the angle (Ψ) of the guided surface wave tilt atR_(x). The position of the coil tap 224 can be adjusted until theoperating point is reached, resulting in the ground current through theammeter 236 substantially reaching a maximum. The resultant fields aresubstantially mode-matched to a guided surface waveguide mode on thesurface of the lossy conducting medium 203, and a guided surface wave islaunched along the surface of the lossy conducting medium 203. This canbe verified by measuring field strength along a radial extending fromthe guided surface waveguide probe 200. The system may be furtheradjusted to improve coupling by iteratively adjusting the position ofthe tap 227 for the AC source 212 to be at the 50Ω point on the coil 215and adjusting the position of tap 224 and/or 233 to maximize the groundcurrent through the ammeter 236.

Referring back to FIG. 12, operation of a guided surface waveguide probe200 may be controlled to adjust for variations in operational conditionsassociated with the guided surface waveguide probe 200. For example, aprobe control system 230 can be used to control the feed network 209and/or positioning of the charge terminal T₁ and/or compensationterminal T₂ to control the operation of the guided surface waveguideprobe 200. Operational conditions can include, but are not limited to,variations in the characteristics of the lossy conducting medium 203(e.g., conductivity σ and relative permittivity ε_(r)), variations infield strength and/or variations in loading of the guided surfacewaveguide probe 200. As can be seen from Equations (41)-(44), the indexof refraction (n), the complex Brewster angle (θ_(i,B) and ψ_(i,B)), thewave tilt (|W|e^(jΨ)) and the complex effective height(h_(eff)=h_(p)e^(jΦ)) can be affected by changes in soil conductivityand permittivity resulting from, e.g., weather conditions.

Equipment such as, e.g., conductivity measurement probes, permittivitysensors, ground parameter meters, field meters, current monitors and/orload receivers can be used to monitor for changes in the operationalconditions and provide information about current operational conditionsto the probe control system 230. The probe control system 230 can thenmake one or more adjustments to the guided surface waveguide probe 200to maintain specified operational conditions for the guided surfacewaveguide probe 200. For instance, as the moisture and temperature vary,the conductivity of the soil will also vary. Conductivity measurementprobes and/or permittivity sensors may be located at multiple locationsaround the guided surface waveguide probe 200. Generally, it would bedesirable to monitor the conductivity and/or permittivity at or aboutthe Hankel crossover distance R_(x) for the operational frequency.Conductivity measurement probes and/or permittivity sensors may belocated at multiple locations (e.g., in each quadrant) around the guidedsurface waveguide probe 200.

With reference then to FIG. 16, shown is an example of a guided surfacewaveguide probe 200 e that includes a charge terminal T₁ and a chargeterminal T₂ that are arranged along a vertical axis z. The guidedsurface waveguide probe 200 e is disposed above a lossy conductingmedium 203, which makes up Region 1. In addition, a second medium 206shares a boundary interface with the lossy conducting medium 203 andmakes up Region 2. The charge terminals T₁ and T₂ are positioned overthe lossy conducting medium 203. The charge terminal T₁ is positioned atheight H₁, and the charge terminal T₂ is positioned directly below T₁along the vertical axis z at height H₂, where H₂ is less than H₁. Theheight h of the transmission structure presented by the guided surfacewaveguide probe 200 e is h=H₁−H₂. The guided surface waveguide probe 200e includes a probe feed network 209 that couples an excitation source212 to the charge terminals T₁ and T₂.

The charge terminals T₁ and/or T₂ include a conductive mass that canhold an electrical charge, which may be sized to hold as much charge aspractically possible. The charge terminal T₁ has a self-capacitance C₁,and the charge terminal T₂ has a self-capacitance C₂, which can bedetermined using, for example, equation (24). By virtue of the placementof the charge terminal T₁ directly above the charge terminal T₂, amutual capacitance C_(M) is created between the charge terminals T₁ andT₂. Note that the charge terminals T₁ and T₂ need not be identical, buteach can have a separate size and shape, and can include differentconducting materials. Ultimately, the field strength of a guided surfacewave launched by a guided surface waveguide probe 200 e is directlyproportional to the quantity of charge on the terminal T₁. The charge Q₁is, in turn, proportional to the self-capacitance C₁ associated with thecharge terminal T₁ since Q₁=C₁V, where V is the voltage imposed on thecharge terminal T₁.

When properly adjusted to operate at a predefined operating frequency,the guided surface waveguide probe 200 e generates a guided surface wavealong the surface of the lossy conducting medium 203. The excitationsource 212 can generate electrical energy at the predefined frequencythat is applied to the guided surface waveguide probe 200 e to excitethe structure. When the electromagnetic fields generated by the guidedsurface waveguide probe 200 e are substantially mode-matched with thelossy conducting medium 203, the electromagnetic fields substantiallysynthesize a wave front incident at a complex Brewster angle thatresults in little or no reflection. Thus, the surface waveguide probe200 e does not produce a radiated wave, but launches a guided surfacetraveling wave along the surface of a lossy conducting medium 203. Theenergy from the excitation source 212 can be transmitted as Zennecksurface currents to one or more receivers that are located within aneffective transmission range of the guided surface waveguide probe 200e.

One can determine asymptotes of the radial Zenneck surface currentJ_(ρ)(ρ) on the surface of the lossy conducting medium 203 to be J₁(ρ)close-in and J₂ (ρ) far-out, where

$\begin{matrix}{{{{{Close}\text{-}{in}\mspace{14mu}\left( {\rho < {N\; 8}} \right)\text{:}\mspace{11mu}{J_{\rho}(\rho)}} \sim J_{1}} = {\frac{l_{1} + l_{2}}{2\pi\;\rho} + \frac{{E_{\rho}^{QS}\left( Q_{1} \right)} + {E_{\rho}^{QS}\left( Q_{2} \right)}}{Z_{\rho}}}},{and}} & (90) \\{{{{Far}\text{-}{out}\mspace{14mu}\left( {\rho\operatorname{>>}{N\; 8}} \right)\text{:}\mspace{11mu}{J_{\rho}(\rho)}} \sim J_{2}} = {\frac{j\;\gamma\;\omega\; Q_{1}}{4} \times \sqrt{\frac{2\gamma}{\pi}} \times {\frac{e^{{- {({\alpha + {j\;\beta}})}}\rho}}{\sqrt{\rho}}.}}} & (91)\end{matrix}$where I₁ is the conduction current feeding the charge Q₁ on the firstcharge terminal T₁, and I₂ is the conduction current feeding the chargeQ₂ on the second charge terminal T₂. The charge Q₁ on the upper chargeterminal T₁ is determined by Q₁=C₁V₁, where C₁ is the isolatedcapacitance of the charge terminal T₁. Note that there is a thirdcomponent to J₁ set forth above given by (E_(ρ) ^(Q) ¹ )/Z_(ρ), whichfollows from the Leontovich boundary condition and is the radial currentcontribution in the lossy conducting medium 203 pumped by thequasi-static field of the elevated oscillating charge on the firstcharge terminal Q₁. The quantity Z_(ρ)=jωμ_(o)/γ_(e) is the radialimpedance of the lossy conducting medium, whereγ_(e)=(Jωμ₁σ₁−ω²μ₁ε₁)^(1/2).

The asymptotes representing the radial current close-in and far-out asset forth by equations (90) and (91) are complex quantities. Accordingto various embodiments, a physical surface current J(ρ), is synthesizedto match as close as possible the current asymptotes in magnitude andphase. That is to say close-in, |J(ρ)| is to be tangent to |J₁|, andfar-out |J(ρ)| is to be tangent to |J₂|. Also, according to the variousembodiments, the phase of J(ρ) should transition from the phase of J₁close-in to the phase of J₂ far-out.

In order to match the guided surface wave mode at the site oftransmission to launch a guided surface wave, the phase of the surfacecurrent |J₂| far-out should differ from the phase of the surface current|J₁| close-in by the propagation phase corresponding to e^(−jβ(ρ) ² ^(ρ)¹ ⁾ plus a constant of approximately 45 degrees or 225 degrees. This isbecause there are two roots for √{square root over (γ)}, one near π/4and one near 5π/4. The properly adjusted synthetic radial surfacecurrent is

$\begin{matrix}{{J_{\rho}\left( {\rho,\phi,0} \right)} = {\frac{I_{o}\gamma}{4}{{H_{1}^{(2)}\left( {{- j}\;\gamma\;\rho} \right)}.}}} & (92)\end{matrix}$Note that this is consistent with equation (17). By Maxwell's equations,such a J(ρ) surface current automatically creates fields that conform to

$\begin{matrix}{{H_{\phi} = {\frac{{- \gamma}\; l_{o}}{4}e^{{- u_{2}}z}{H_{1}^{(2)}\left( {{- j}\;\gamma\;\rho} \right)}}},} & (93) \\{{E_{\rho} = {\frac{{- \gamma}\; l_{o}}{4}\left( \frac{u_{2}}{j\;\omega\; ɛ_{o}} \right)e^{{- u_{2}}z}{H_{1}^{(2)}\left( {{- j}\;\gamma\;\rho} \right)}}},{and}} & (94) \\{E_{z} = {\frac{{- \gamma}\; l_{o}}{4}\left( \frac{- \gamma}{\omega\; ɛ_{o}} \right)e^{{- u_{2}}z}{{H_{0}^{(2)}\left( {{- j}\;\gamma\;\rho} \right)}.}}} & (95)\end{matrix}$Thus, the difference in phase between the surface current |J₂| far-outand the surface current |J₁| close-in for the guided surface wave modethat is to be matched is due to the characteristics of the Hankelfunctions in equations (93)-(95), which are consistent with equations(1)-(3). It is of significance to recognize that the fields expressed byequations (1)-(6) and (17) and equations (92)-(95) have the nature of atransmission line mode bound to a lossy interface, not radiation fieldsthat are associated with groundwave propagation.

In order to obtain the appropriate voltage magnitudes and phases for agiven design of a guided surface waveguide probe 200 e at a givenlocation, an iterative approach may be used. Specifically, analysis maybe performed of a given excitation and configuration of a guided surfacewaveguide probe 200 e taking into account the feed currents to theterminals T₁ and T₂, the charges on the charge terminals T₁ and T₂, andtheir images in the lossy conducting medium 203 in order to determinethe radial surface current density generated. This process may beperformed iteratively until an optimal configuration and excitation fora given guided surface waveguide probe 200 e is determined based ondesired parameters. To aid in determining whether a given guided surfacewaveguide probe 200 e is operating at an optimal level, a guided fieldstrength curve 103 (FIG. 1) may be generated using equations (1)-(12)based on values for the conductivity of Region 1 (σ₁) and thepermittivity of Region 1 (ε₁) at the location of the guided surfacewaveguide probe 200 e. Such a guided field strength curve 103 canprovide a benchmark for operation such that measured field strengths canbe compared with the magnitudes indicated by the guided field strengthcurve 103 to determine if optimal transmission has been achieved.

In order to arrive at an optimized condition, various parametersassociated with the guided surface waveguide probe 200 e may beadjusted. One parameter that may be varied to adjust the guided surfacewaveguide probe 200 e is the height of one or both of the chargeterminals T₁ and/or T₂ relative to the surface of the lossy conductingmedium 203. In addition, the distance or spacing between the chargeterminals T₁ and T₂ may also be adjusted. In doing so, one may minimizeor otherwise alter the mutual capacitance C_(M) or any boundcapacitances between the charge terminals T₁ and T₂ and the lossyconducting medium 203 as can be appreciated. The size of the respectivecharge terminals T₁ and/or T₂ can also be adjusted. By changing the sizeof the charge terminals T₁ and/or T₂, one will alter the respectiveself-capacitances C₁ and/or C₂, and the mutual capacitance C_(M) as canbe appreciated.

Still further, another parameter that can be adjusted is the probe feednetwork 209 associated with the guided surface waveguide probe 200 e.This may be accomplished by adjusting the size of the inductive and/orcapacitive reactances that make up the probe feed network 209. Forexample, where such inductive reactances comprise coils, the number ofturns on such coils may be adjusted. Ultimately, the adjustments to theprobe feed network 209 can be made to alter the electrical length of theprobe feed network 209, thereby affecting the voltage magnitudes andphases on the charge terminals T₁ and T₂.

Note that the iterations of transmission performed by making the variousadjustments may be implemented by using computer models or by adjustingphysical structures as can be appreciated. By making the aboveadjustments, one can create corresponding “close-in” surface current J₁and “far-out” surface current J₂ that approximate the same currents J(ρ)of the guided surface wave mode specified in Equations (90) and (91) setforth above. In doing so, the resulting electromagnetic fields would besubstantially or approximately mode-matched to a guided surface wavemode on the surface of the lossy conducting medium 203.

While not shown in the example of FIG. 16, operation of the guidedsurface waveguide probe 200 e may be controlled to adjust for variationsin operational conditions associated with the guided surface waveguideprobe 200. For example, a probe control system 230 shown in FIG. 12 canbe used to control the feed network 209 and/or positioning and/or sizeof the charge terminals T₁ and/or T₂ to control the operation of theguided surface waveguide probe 200 e. Operational conditions caninclude, but are not limited to, variations in the characteristics ofthe lossy conducting medium 203 (e.g., conductivity σ and relativepermittivity ε_(r)), variations in field strength and/or variations inloading of the guided surface waveguide probe 200 e.

Referring now to FIG. 17, shown is an example of the guided surfacewaveguide probe 200 e of FIG. 16, denoted herein as guided surfacewaveguide probe 200 f. The guided surface waveguide probe 200 f includesthe charge terminals T₁ and T₂ that are positioned along a vertical axisz that is substantially normal to the plane presented by the lossyconducting medium 203 (e.g., the Earth). The second medium 206 is abovethe lossy conducting medium 203. The charge terminal T₁ has aself-capacitance C₁, and the charge terminal T₂ has a self-capacitanceC₂. During operation, charges Q₁ and Q₂ are imposed on the chargeterminals T₁ and T₂, respectively, depending on the voltages applied tothe charge terminals T₁ and T₂ at any given instant. A mutualcapacitance C_(M) may exist between the charge terminals T₁ and T₂depending on the distance there between. In addition, bound capacitancesmay exist between the respective charge terminals T₁ and T₂ and thelossy conducting medium 203 depending on the heights of the respectivecharge terminals T₁ and T₂ with respect to the lossy conducting medium203.

The guided surface waveguide probe 200 f includes a probe feed network209 that comprises an inductive impedance comprising a coil L_(1a)having a pair of leads that are coupled to respective ones of the chargeterminals T₁ and T₂. In one embodiment, the coil L_(1a) is specified tohave an electrical length that is one-half (½) of the wavelength at theoperating frequency of the guided surface waveguide probe 200 f.

While the electrical length of the coil L_(1a) is specified asapproximately one-half (½) the wavelength at the operating frequency, itis understood that the coil L_(1a) may be specified with an electricallength at other values. According to one embodiment, the fact that thecoil L_(1a) has an electrical length of approximately one-half thewavelength at the operating frequency provides for an advantage in thata maximum voltage differential is created on the charge terminals T₁ andT₂. Nonetheless, the length or diameter of the coil L_(1a) may beincreased or decreased when adjusting the guided surface waveguide probe200 f to obtain optimal excitation of a guided surface wave mode.Adjustment of the coil length may be provided by taps located at one orboth ends of the coil. In other embodiments, it may be the case that theinductive impedance is specified to have an electrical length that issignificantly less than or greater than ½ the wavelength at theoperating frequency of the guided surface waveguide probe 200 f.

The excitation source 212 can be coupled to the probe feed network 209by way of magnetic coupling. Specifically, the excitation source 212 iscoupled to a coil L_(P) that is inductively coupled to the coil L_(1a).This may be done by link coupling, a tapped coil, a variable reactance,or other coupling approach as can be appreciated. To this end, the coilL_(P) acts as a primary, and the coil L_(1a) acts as a secondary as canbe appreciated.

In order to adjust the guided surface waveguide probe 200 f for thetransmission of a desired guided surface wave, the heights of therespective charge terminals T₁ and T₂ may be altered with respect to thelossy conducting medium 203 and with respect to each other. Also, thesizes of the charge terminals T₁ and T₂ may be altered. In addition, thesize of the coil L_(1a) may be altered by adding or eliminating turns orby changing some other dimension of the coil L_(1a). The coil L_(1a) canalso include one or more taps for adjusting the electrical length asshown in FIG. 17. The position of a tap connected to either chargeterminal T₁ or T₂ can also be adjusted.

Referring next to FIGS. 18A, 18B, 18C and 19, shown are examples ofgeneralized receive circuits for using the surface-guided waves inwireless power delivery systems. FIGS. 18A and 18B-18C include a linearprobe 303 and a tuned resonator 306, respectively. FIG. 19 is a magneticcoil 309 according to various embodiments of the present disclosure.According to various embodiments, each one of the linear probe 303, thetuned resonator 306, and the magnetic coil 309 may be employed toreceive power transmitted in the form of a guided surface wave on thesurface of a lossy conducting medium 203 according to variousembodiments. As mentioned above, in one embodiment the lossy conductingmedium 203 comprises a terrestrial medium (or Earth).

With specific reference to FIG. 18A, the open-circuit terminal voltageat the output terminals 312 of the linear probe 303 depends upon theeffective height of the linear probe 303. To this end, the terminalpoint voltage may be calculated asV _(T)=∫₀ ^(h) ^(e) E _(inc) ·dl,  (96)where E_(inc) is the strength of the incident electric field induced onthe linear probe 303 in Volts per meter, dl is an element of integrationalong the direction of the linear probe 303, and h_(e) is the effectiveheight of the linear probe 303. An electrical load 315 is coupled to theoutput terminals 312 through an impedance matching network 318.

When the linear probe 303 is subjected to a guided surface wave asdescribed above, a voltage is developed across the output terminals 312that may be applied to the electrical load 315 through a conjugateimpedance matching network 318 as the case may be. In order tofacilitate the flow of power to the electrical load 315, the electricalload 315 should be substantially impedance matched to the linear probe303 as will be described below.

Referring to FIG. 18B, a ground current excited coil 306 a possessing aphase shift equal to the wave tilt of the guided surface wave includes acharge terminal T_(R) that is elevated (or suspended) above the lossyconducting medium 203. The charge terminal T_(R) has a self-capacitanceC_(R). In addition, there may also be a bound capacitance (not shown)between the charge terminal T_(R) and the lossy conducting medium 203depending on the height of the charge terminal T_(R) above the lossyconducting medium 203. The bound capacitance should preferably beminimized as much as is practicable, although this may not be entirelynecessary in every instance.

The tuned resonator 306 a also includes a receiver network comprising acoil L_(R) having a phase shift Φ. One end of the coil L_(R) is coupledto the charge terminal T_(R), and the other end of the coil L_(R) iscoupled to the lossy conducting medium 203. The receiver network caninclude a vertical supply line conductor that couples the coil L_(R) tothe charge terminal T_(R). To this end, the coil L_(R) (which may alsobe referred to as tuned resonator L_(R)-C_(R)) comprises aseries-adjusted resonator as the charge terminal C_(R) and the coilL_(R) are situated in series. The phase delay of the coil L_(R) can beadjusted by changing the size and/or height of the charge terminalT_(R), and/or adjusting the size of the coil L_(R) so that the phase Φof the structure is made substantially equal to the angle of the wavetilt Ψ. The phase delay of the vertical supply line can also be adjustedby, e.g., changing length of the conductor.

For example, the reactance presented by the self-capacitance C_(R) iscalculated as 1/jωC_(R). Note that the total capacitance of thestructure 306 a may also include capacitance between the charge terminalT_(R) and the lossy conducting medium 203, where the total capacitanceof the structure 306 a may be calculated from both the self-capacitanceC_(R) and any bound capacitance as can be appreciated. According to oneembodiment, the charge terminal T_(R) may be raised to a height so as tosubstantially reduce or eliminate any bound capacitance. The existenceof a bound capacitance may be determined from capacitance measurementsbetween the charge terminal T_(R) and the lossy conducting medium 203 aspreviously discussed.

The inductive reactance presented by a discrete-element coil L_(R) maybe calculated as jωL, where L is the lumped-element inductance of thecoil L_(R). If the coil L_(R) is a distributed element, its equivalentterminal-point inductive reactance may be determined by conventionalapproaches. To tune the structure 306 a, one would make adjustments sothat the phase delay is equal to the wave tilt for the purpose ofmode-matching to the surface waveguide at the frequency of operation.Under this condition, the receiving structure may be considered to be“mode-matched” with the surface waveguide. A transformer link around thestructure and/or an impedance matching network 324 may be insertedbetween the probe and the electrical load 327 in order to couple powerto the load. Inserting the impedance matching network 324 between theprobe terminals 321 and the electrical load 327 can effect aconjugate-match condition for maximum power transfer to the electricalload 327.

When placed in the presence of surface currents at the operatingfrequencies power will be delivered from the surface guided wave to theelectrical load 327. To this end, an electrical load 327 may be coupledto the structure 306 a by way of magnetic coupling, capacitive coupling,or conductive (direct tap) coupling. The elements of the couplingnetwork may be lumped components or distributed elements as can beappreciated.

In the embodiment shown in FIG. 18B, magnetic coupling is employed wherea coil L_(S) is positioned as a secondary relative to the coil L_(R)that acts as a transformer primary. The coil L_(S) may be link-coupledto the coil L_(R) by geometrically winding it around the same corestructure and adjusting the coupled magnetic flux as can be appreciated.In addition, while the receiving structure 306 a comprises aseries-tuned resonator, a parallel-tuned resonator or even adistributed-element resonator of the appropriate phase delay may also beused.

While a receiving structure immersed in an electromagnetic field maycouple energy from the field, it can be appreciated thatpolarization-matched structures work best by maximizing the coupling,and conventional rules for probe-coupling to waveguide modes should beobserved. For example, a TE₂₀ (transverse electric mode) waveguide probemay be optimal for extracting energy from a conventional waveguideexcited in the TE₂₀ mode. Similarly, in these cases, a mode-matched andphase-matched receiving structure can be optimized for coupling powerfrom a surface-guided wave. The guided surface wave excited by a guidedsurface waveguide probe 200 on the surface of the lossy conductingmedium 203 can be considered a waveguide mode of an open waveguide.Excluding waveguide losses, the source energy can be completelyrecovered. Useful receiving structures may be E-field coupled, H-fieldcoupled, or surface-current excited.

The receiving structure can be adjusted to increase or maximize couplingwith the guided surface wave based upon the local characteristics of thelossy conducting medium 203 in the vicinity of the receiving structure.To accomplish this, the phase delay (Φ) of the receiving structure canbe adjusted to match the angle (Ψ) of the wave tilt of the surfacetraveling wave at the receiving structure. If configured appropriately,the receiving structure may then be tuned for resonance with respect tothe perfectly conducting image ground plane at complex depth z=−d/2.

For example, consider a receiving structure comprising the tunedresonator 306 a of FIG. 18B, including a coil L_(R) and a verticalsupply line connected between the coil L_(R) and a charge terminalT_(R). With the charge terminal T_(R) positioned at a defined heightabove the lossy conducting medium 203, the total phase shift (of thecoil L_(R) and vertical supply line can be matched with the angle (Ψ) ofthe wave tilt at the location of the tuned resonator 306 a. FromEquation (22), it can be seen that the wave tilt asymptotically passesto

$\begin{matrix}{{W = {{{W}e^{j\;\Psi}} = {\frac{E_{\rho}}{E_{z}}\underset{\;{\rho\rightarrow\infty}\mspace{14mu}}{\rightarrow}\frac{1}{\sqrt{ɛ_{r} - {j\frac{\sigma_{1}}{\omega\; ɛ_{o}}}}}}}},} & (97)\end{matrix}$where ε_(r) comprises the relative permittivity and σ₁ is theconductivity of the lossy conducting medium 203 at the location of thereceiving structure, ε_(o) is the permittivity of free space, and ω=2πf,where f is the frequency of excitation. Thus, the wave tilt angle (Ψ)can be determined from Equation (97).

The total phase shift (Φ=θ_(c)+θ_(y)) of the tuned resonator 306 aincludes both the phase delay (θ_(c)) through the coil L_(R) and thephase delay of the vertical supply line (θ_(y)). The spatial phase delayalong the conductor length l_(w) of the vertical supply line can begiven by θ_(y)=β_(w)l_(w), where β_(w) is the propagation phase constantfor the vertical supply line conductor. The phase delay due to the coil(or helical delay line) is θ_(c)=β_(p)l_(C), with a physical length ofl_(C) and a propagation factor of

$\begin{matrix}{{\beta_{p} = {\frac{2\pi}{\lambda_{p}} = \frac{2\pi}{V_{f}\lambda_{0}}}},} & (98)\end{matrix}$where V_(f) is the velocity factor on the structure, λ₀ is thewavelength at the supplied frequency, and λ_(p) is the propagationwavelength resulting from the velocity factor V_(f). One or both of thephase delays (θ_(c)+θ_(y)) can be adjusted to match the phase shift Φ tothe angle (Ψ) of the wave tilt. For example, a tap position may beadjusted on the coil L_(R) of FIG. 18B to adjust the coil phase delay(θ_(c)) to match the total phase shift to the wave tilt angle (Φ=Ψ). Forexample, a portion of the coil can be bypassed by the tap connection asillustrated in FIG. 18B. The vertical supply line conductor can also beconnected to the coil L_(R) via a tap, whose position on the coil may beadjusted to match the total phase shift to the angle of the wave tilt.

Once the phase delay (Φ) of the tuned resonator 306 a has been adjusted,the impedance of the charge terminal T_(R) can then be adjusted to tuneto resonance with respect to the perfectly conducting image ground planeat complex depth z=−d/2. This can be accomplished by adjusting thecapacitance of the charge terminal T₁ without changing the travelingwave phase delays of the coil L_(R) and vertical supply line. Theadjustments are similar to those described with respect to FIGS. 9A and9B.

The impedance seen “looking down” into the lossy conducting medium 203to the complex image plane is given by:Z _(in) =R _(in) +jX _(in) =Z _(o) tan h(jβ _(o)(d/2)),  (99)where β_(o)=ω√{square root over (μ_(o)ε_(o))}. For vertically polarizedsources over the Earth, the depth of the complex image plane can begiven by:d/2≈1/√{square root over (jωμ ₁σ₁−ω²μ₁ε₁)},  (100)where μ₁ is the permeability of the lossy conducting medium 203 andε₁=ε_(r)ε_(o).

At the base of the tuned resonator 306 a, the impedance seen “lookingup” into the receiving structure is Z_(↑)=Z_(base) as illustrated inFIG. 9A. With a terminal impedance of:

$\begin{matrix}{{Z_{R} = \frac{1}{j\;\omega\; C_{R}}},} & (101)\end{matrix}$where C_(R) is the self-capacitance of the charge terminal T_(R), theimpedance seen “looking up” into the vertical supply line conductor ofthe tuned resonator 306 a is given by:

$\begin{matrix}{{Z_{2} = {{Z_{W}\frac{Z_{R} + {Z_{W}{\tanh\left( {j\;\beta_{W}h_{W}} \right)}}}{Z_{W} + {Z_{R}{\tanh\left( {j\;\beta_{W}h_{W}} \right)}}}} = {Z_{W}\frac{Z_{R} + {Z_{W}{\tanh\left( {j\;\theta_{y}} \right)}}}{Z_{W} + {Z_{R}{\tanh\left( {j\;\theta_{y}} \right)}}}}}},} & (102)\end{matrix}$and the impedance seen “looking up” into the coil L_(R) of the tunedresonator 306 a is given by:

$\begin{matrix}{Z_{base} = {{R_{base} + {j\; X_{base}}} = {{Z_{R}\frac{Z_{2} + {Z_{R}{\tanh\left( {j\;\beta_{p}H} \right)}}}{Z_{R} + {Z_{2}{\tanh\left( {j\;\beta_{p}H} \right)}}}} = {Z_{c}{\frac{Z_{2} + {Z_{R}{\tanh\left( {j\;\theta_{c}} \right)}}}{Z_{R} + {Z_{2}{\tanh\left( {j\;\theta_{c}} \right)}}}.}}}}} & (103)\end{matrix}$By matching the reactive component (X_(in)) seen “looking down” into thelossy conducting medium 203 with the reactive component (X_(base)) seen“looking up” into the tuned resonator 306 a, the coupling into theguided surface waveguide mode may be maximized.

Referring next to FIG. 18C, shown is an example of a tuned resonator 306b that does not include a charge terminal T_(R) at the top of thereceiving structure. In this embodiment, the tuned resonator 306 b doesnot include a vertical supply line coupled between the coil L_(R) andthe charge terminal T_(R). Thus, the total phase shift (Φ) of the tunedresonator 306 b includes only the phase delay (θ_(c)) through the coilL_(R). As with the tuned resonator 306 a of FIG. 18B, the coil phasedelay θ_(c) can be adjusted to match the angle (Ψ) of the wave tiltdetermined from Equation (97), which results in Φ=Ψ. While powerextraction is possible with the receiving structure coupled into thesurface waveguide mode, it is difficult to adjust the receivingstructure to maximize coupling with the guided surface wave without thevariable reactive load provided by the charge terminal T_(R).

Referring to FIG. 18D, shown is a flow chart 180 illustrating an exampleof adjusting a receiving structure to substantially mode-match to aguided surface waveguide mode on the surface of the lossy conductingmedium 203. Beginning with 181, if the receiving structure includes acharge terminal T_(R) (e.g., of the tuned resonator 306 a of FIG. 18B),then the charge terminal T_(R) is positioned at a defined height above alossy conducting medium 203 at 184. As the surface guided wave has beenestablished by a guided surface waveguide probe 200, the physical height(h_(p)) of the charge terminal T_(R) may be below that of the effectiveheight. The physical height may be selected to reduce or minimize thebound charge on the charge terminal T_(R) (e.g., four times thespherical diameter of the charge terminal). If the receiving structuredoes not include a charge terminal T_(R) (e.g., of the tuned resonator306 b of FIG. 18C), then the flow proceeds to 187.

At 187, the electrical phase delay Φ of the receiving structure ismatched to the complex wave tilt angle Ψ defined by the localcharacteristics of the lossy conducting medium 203. The phase delay(θ_(c)) of the helical coil and/or the phase delay (θ_(y)) of thevertical supply line can be adjusted to make Φ equal to the angle (Ψ) ofthe wave tilt (W). The angle (Ψ) of the wave tilt can be determined fromEquation (86). The electrical phase Φ can then be matched to the angleof the wave tilt. For example, the electrical phase delay Φ=θ_(c)+θ_(y)can be adjusted by varying the geometrical parameters of the coil L_(R)and/or the length (or height) of the vertical supply line conductor.

Next at 190, the load impedance of the charge terminal T_(R) can betuned to resonate the equivalent image plane model of the tunedresonator 306 a. The depth (d/2) of the conducting image ground plane139 (FIG. 9A) below the receiving structure can be determined usingEquation (100) and the values of the lossy conducting medium 203 (e.g.,the Earth) at the receiving structure, which can be locally measured.Using that complex depth, the phase shift (θ_(d)) between the imageground plane 139 and the physical boundary 136 (FIG. 9A) of the lossyconducting medium 203 can be determined using θ_(d)=β_(o)d/2. Theimpedance (Z_(in)) as seen “looking down” into the lossy conductingmedium 203 can then be determined using Equation (99). This resonancerelationship can be considered to maximize coupling with the guidedsurface waves.

Based upon the adjusted parameters of the coil L_(R) and the length ofthe vertical supply line conductor, the velocity factor, phase delay,and impedance of the coil L_(R) and vertical supply line can bedetermined. In addition, the self-capacitance (C_(R)) of the chargeterminal T_(R) can be determined using, e.g., Equation (24). Thepropagation factor (β_(p)) of the coil L_(R) can be determined usingEquation (98), and the propagation phase constant (β_(w)) for thevertical supply line can be determined using Equation (49). Using theself-capacitance and the determined values of the coil L_(R) andvertical supply line, the impedance (Z_(base)) of the tuned resonator306 a as seen “looking up” into the coil L_(R) can be determined usingEquations (101), (102), and (103).

The equivalent image plane model of FIG. 9A also applies to the tunedresonator 306 a of FIG. 18B. The tuned resonator 306 a can be tuned toresonance with respect to the complex image plane by adjusting the loadimpedance Z_(R) of the charge terminal T_(R) such that the reactancecomponent X_(base) of Z_(base) cancels out the reactance component ofX_(in) of Z_(in), or X_(base)+X_(in)=0. Thus, the impedance at thephysical boundary 136 (FIG. 9A) “looking up” into the coil of the tunedresonator 306 a is the conjugate of the impedance at the physicalboundary 136 “looking down” into the lossy conducting medium 203. Theload impedance Z_(R) can be adjusted by varying the capacitance (C_(R))of the charge terminal T_(R) without changing the electrical phase delayΦ=θ_(c)+θ_(y) seen by the charge terminal T_(R). An iterative approachmay be taken to tune the load impedance Z_(R) for resonance of theequivalent image plane model with respect to the conducting image groundplane 139. In this way, the coupling of the electric field to a guidedsurface waveguide mode along the surface of the lossy conducting medium203 (e.g., Earth) can be improved and/or maximized.

Referring to FIG. 19, the magnetic coil 309 comprises a receive circuitthat is coupled through an impedance matching network 333 to anelectrical load 336. In order to facilitate reception and/or extractionof electrical power from a guided surface wave, the magnetic coil 309may be positioned so that the magnetic flux of the guided surface wave,H_(φ), passes through the magnetic coil 309, thereby inducing a currentin the magnetic coil 309 and producing a terminal point voltage at itsoutput terminals 330. The magnetic flux of the guided surface wavecoupled to a single turn coil is expressed by

=∫∫_(A) _(CS) μ_(r)μ_(o)

·{circumflex over (n)}dA  (104)where

is the coupled magnetic flux, μ_(r) is the effective relativepermeability of the core of the magnetic coil 309, μ_(o) is thepermeability of free space,

is the incident magnetic field strength vector, {circumflex over (n)} isa unit vector normal to the cross-sectional area of the turns, andA_(CS) is the area enclosed by each loop. For an N-turn magnetic coil309 oriented for maximum coupling to an incident magnetic field that isuniform over the cross-sectional area of the magnetic coil 309, theopen-circuit induced voltage appearing at the output terminals 330 ofthe magnetic coil 309 is

$\begin{matrix}{{V = {{{- N}\frac{d\;\mathcal{F}}{d\; t}} \approx {{- j}\;\omega\;\mu_{r}\mu_{0}{NHA}_{CS}}}},} & (105)\end{matrix}$where the variables are defined above. The magnetic coil 309 may betuned to the guided surface wave frequency either as a distributedresonator or with an external capacitor across its output terminals 330,as the case may be, and then impedance-matched to an external electricalload 336 through a conjugate impedance matching network 333.

Assuming that the resulting circuit presented by the magnetic coil 309and the electrical load 336 are properly adjusted and conjugateimpedance matched, via impedance matching network 333, then the currentinduced in the magnetic coil 309 may be employed to optimally power theelectrical load 336. The receive circuit presented by the magnetic coil309 provides an advantage in that it does not have to be physicallyconnected to the ground.

With reference to FIGS. 18A, 18B, 18C and 19, the receive circuitspresented by the linear probe 303, the mode-matched structure 306, andthe magnetic coil 309 each facilitate receiving electrical powertransmitted from any one of the embodiments of guided surface waveguideprobes 200 described above. To this end, the energy received may be usedto supply power to an electrical load 315/327/336 via a conjugatematching network as can be appreciated. This contrasts with the signalsthat may be received in a receiver that were transmitted in the form ofa radiated electromagnetic field. Such signals have very low availablepower, and receivers of such signals do not load the transmitters.

It is also characteristic of the present guided surface waves generatedusing the guided surface waveguide probes 200 described above that thereceive circuits presented by the linear probe 303, the mode-matchedstructure 306, and the magnetic coil 309 will load the excitation source212 (e.g., FIGS. 3, 12 and 16) that is applied to the guided surfacewaveguide probe 200, thereby generating the guided surface wave to whichsuch receive circuits are subjected. This reflects the fact that theguided surface wave generated by a given guided surface waveguide probe200 described above comprises a transmission line mode. By way ofcontrast, a power source that drives a radiating antenna that generatesa radiated electromagnetic wave is not loaded by the receivers,regardless of the number of receivers employed.

Thus, together one or more guided surface waveguide probes 200 and oneor more receive circuits in the form of the linear probe 303, the tunedmode-matched structure 306, and/or the magnetic coil 309 can make up awireless distribution system. Given that the distance of transmission ofa guided surface wave using a guided surface waveguide probe 200 as setforth above depends upon the frequency, it is possible that wirelesspower distribution can be achieved across wide areas and even globally.

The conventional wireless-power transmission/distribution systemsextensively investigated today include “energy harvesting” fromradiation fields and also sensor coupling to inductive or reactivenear-fields. In contrast, the present wireless-power system does notwaste power in the form of radiation which, if not intercepted, is lostforever. Nor is the presently disclosed wireless-power system limited toextremely short ranges as with conventional mutual-reactance couplednear-field systems. The wireless-power system disclosed hereinprobe-couples to the novel surface-guided transmission line mode, whichis equivalent to delivering power to a load by a waveguide or a loaddirectly wired to the distant power generator. Not counting the powerrequired to maintain transmission field strength plus that dissipated inthe surface waveguide, which at extremely low frequencies isinsignificant relative to the transmission losses in conventionalhigh-tension power lines at 60 Hz, all of the generator power goes onlyto the desired electrical load. When the electrical load demand isterminated, the source power generation is relatively idle.

Referring next to FIGS. 20A-E, shown are examples of various schematicsymbols that are used with reference to the discussion that follows.With specific reference to FIG. 20A, shown is a symbol that representsany one of the guided surface waveguide probes 200 a, 200 b, 200 c, 200e, 200 d, or 200 f; or any variations thereof. In the following drawingsand discussion, a depiction of this symbol will be referred to as aguided surface waveguide probe P. For the sake of simplicity in thefollowing discussion, any reference to the guided surface waveguideprobe P is a reference to any one of the guided surface waveguide probes200 a, 200 b, 200 c, 200 e, 200 d, or 200 f; or variations thereof.

Similarly, with reference to FIG. 20B, shown is a symbol that representsa guided surface wave receive structure that may comprise any one of thelinear probe 303 (FIG. 18A), the tuned resonator 306 (FIGS. 18B-18C), orthe magnetic coil 309 (FIG. 19). In the following drawings anddiscussion, a depiction of this symbol will be referred to as a guidedsurface wave receive structure R. For the sake of simplicity in thefollowing discussion, any reference to the guided surface wave receivestructure R is a reference to any one of the linear probe 303, the tunedresonator 306, or the magnetic coil 309; or variations thereof.

Further, with reference to FIG. 20C, shown is a symbol that specificallyrepresents the linear probe 303 (FIG. 18A). In the following drawingsand discussion, a depiction of this symbol will be referred to as aguided surface wave receive structure R_(P). For the sake of simplicityin the following discussion, any reference to the guided surface wavereceive structure R_(P) is a reference to the linear probe 303 orvariations thereof.

Further, with reference to FIG. 20D, shown is a symbol that specificallyrepresents the tuned resonator 306 (FIGS. 18B-18C). In the followingdrawings and discussion, a depiction of this symbol will be referred toas a guided surface wave receive structure R_(R). For the sake ofsimplicity in the following discussion, any reference to the guidedsurface wave receive structure R_(R) is a reference to the tunedresonator 306 or variations thereof.

Further, with reference to FIG. 20E, shown is a symbol that specificallyrepresents the magnetic coil 309 (FIG. 19). In the following drawingsand discussion, a depiction of this symbol will be referred to as aguided surface wave receive structure R_(M). For the sake of simplicityin the following discussion, any reference to the guided surface wavereceive structure R_(M) is a reference to the magnetic coil 309 orvariations thereof.

Referring next to FIG. 21, shown is an example of a hierarchical powerdistribution system 2100 according to various embodiments. Thehierarchical power distribution system 2100 includes guided surfacewaveguide probes Pa, Pb . . . PN. According to various embodiments, eachof the guided surface waveguide probes Pa . . . N may be employed totransmit electrical energy in the form of a guided surface wave on thesurface of a lossy conducting medium 203 (FIG. 3). Any N number ofguided surface waveguide probes Pa . . . N may be employed to transmitelectrical energy in the form of a guided surface wave to one or moreguided surface wave receive structures R (FIG. 20B), which supply theelectrical energy to one or more electrical loads.

According to various embodiments, each of the guided surface waveguideprobes Pa . . . N may transmit electrical energy at frequenciesassociated with one or more respective frequency sets 2103 a, 2103 b . .. 2103N on a transmission spectrum 2109. In some embodiments, each ofthe guided surface waveguide probes Pa . . . N transmits electricalenergy at frequencies within one of the respective frequency sets 2103 a. . . N on the transmission spectrum 2109. The frequency sets 2103 a . .. N may comprise frequencies that fall within respective frequency bandsor may correspond to sets of discrete frequencies that do not fallwithin a single frequency band. Accordingly, each of the guided surfacewaveguide probes Pa . . . N may transmit electrical energy in the formof a guided surface wave at frequencies within a unique frequency setrelative to other frequency sets 2103 a . . . N. In alternativeembodiments, each of the guided surface waveguide probes Pa . . . N maytransmit electrical energy at frequencies within more than one of therespective frequency sets 2103 a . . . N on the transmission spectrum2109. In some embodiments, each of the guided surface waveguide probesPa . . . N may transmit electrical energy at one or more discretefrequencies within one of the respective frequency sets 2103 a . . . N.In other words, each of the guided surface waveguide probes Pa . . . Nmay transmit within a narrow frequency set within one of the respectivefrequency sets 2103 a . . . N. Accordingly, each of the guided surfacewaveguide probes Pa . . . N may be configured to transmit electricalenergy in the form of a guided surface wave at a unique frequencyrelative to all other frequencies in the transmission spectrum 2109.

Alternatively, multiple the guided surface waveguide probes Pa . . . Nmay be configured to transmit electrical energy at different discretefrequencies (narrow frequency sets) relative to each other within a samefrequency set 2103 a . . . N. Each of the guided surface waveguideprobes Pa . . . N may be configured to transmit electrical energy in oneor more transmission classes 2106 a, 2106 b . . . 2106N. Each of theguided surface waveguide probes Pa . . . N may be assigned to transmitin one or more of the transmission classes 2106 a . . . N based upon oneor more loads to which the electrical energy may be supplied. Forexample, guided surface waveguide probe Pa may transmit electricalenergy that is received by a load designated to receive energytransmitted via the first transmission class 2106 a, while guidedsurface waveguide probe Pb may transmit electrical energy that isreceived by a load designated to receive energy transmitted via thesecond transmission class 2106 b.

Each transmission class 2106 a, 2106 b . . . 2106N corresponds to atleast one of the respective frequency sets 2103 a . . . N comprising atleast one discrete frequency by which each of the guided surfacewaveguide probes Pa . . . N may transmit electrical energy. According tovarious embodiments, each of the transmission classes 2106 a . . . N isassociated with a priority relative to one another for transmittingelectrical energy in the hierarchical power distribution system 2100.The priority may be based upon the one or more loads that are beingsupplied with electrical energy. For example, a load that is designatedto receive energy transmitted by the first transmission class 2106 a maybe classified as being more critical to receive electrical energy than aload designated to receive electrical energy transmitted by the secondtransmission class 2106 b. Thus, in a situation where the hierarchicalpower distribution system 2100 must cease supplying energy to one of theaforementioned loads due to a loss of power generation capacity, thehierarchical power distribution system 2100 may cease supplying energyto the load designated to receive electrical energy transmitted by thesecond transmission class because it is classified as being lesscritical. Alternatively or additionally, the hierarchical powerdistribution system 2100 may reassign one or more guided surface wavereceive structures R that are coupled to the load to a different one ofthe frequency sets 2103 a . . . N to prevent overloading one or morepower sources.

The hierarchical power distribution system 2100 may assign one or morespecific frequencies and/or one or more of the frequency sets 2103 a . .. N to one or more guided surface wave receive structures R to preventoverloading one or more power sources coupled to respective guidedsurface waveguide probes Pa . . . N. The specific frequencies and/or oneor more of the frequency sets 2103 a . . . N may be based upon one ormore loads that need to receive electrical energy. The hierarchicalpower distribution system 2100 may identify a frequency set 2103 a . . .N associated with a load, and accordingly determine one or moreavailable discrete frequencies within the frequency set 2103 a . . . Nto assign to the one or more guided surface wave receive structures R.

The hierarchical power distribution system 2100 may determine that apower source coupled to one of the guided surface waveguide probes Pa .. . N is at risk of being overloaded. Therefore, the hierarchical powerdistribution system 2100 may prevent electrical energy from beingsupplied to one or more of the loads to prevent overloading one or morepower sources. Alternatively or additionally, the hierarchical powerdistribution system 2100 may reassign one or more guided surface wavereceive structures R to a different one of the frequency sets 2103 a . .. N to prevent overloading the one or more power sources. For example,the hierarchical power distribution system 200 may communicate with areceive controller to instruct the one or more guided surface wavereceive structures R to adjust to a frequency other than the one onwhich it is receiving electrical energy. In other embodiments, thehierarchical power distribution system 200 may communicate with acomputer associated with one or more of the guided surface waveguideprobes Pa . . . N and shut down transmission upon detection that a powersource is being overloaded. In embodiments where one guided surfacewaveguide probe P is transmitting electrical energy being supplied tomore than one load, the hierarchical power distribution system 200 maycease transmission of electrical energy to specific loads based upon oneor more characteristics associated with the load. A load may include forexample, medical devices, transportation machines, buildings, or anyother structure that may be supplied with electrical energy. In someembodiments, a range of sizes of the loads determines which of thetransmission classes 2106 a . . . N the loads belong to. The type ofload may also determine which one or more frequencies within one or moreof the respective frequency sets 2103 a . . . N a guided surfacewaveguide probe Pa . . . N will be adjusted to when transmittingelectrical energy.

Referring next to FIG. 22, shown is an example of a transmissionspectrum 2109 which comprises one or more frequencies by which theguided surface waveguide probes Pa . . . N (FIG. 21) may transmitelectrical energy in the form of a guided surface wave according tovarious embodiments. In this example, transmission spectrum 2109includes critical load frequency set 2206, non-critical load frequencyset 2209, and unused frequency sets 2203 a, 2203 b, and 2203 c.According to various embodiments, loads may be classified as eithercritical or non-critical. Thus, to supply electrical energy to one ormore loads that are classified as critical, a guided surface waveguideprobe Pa . . . N may transmit electrical energy at one or morefrequencies within the critical load frequency set 2206. For example, aguided surface waveguide probe Pa . . . N may transmit at a discretefrequency 2212 within the critical load frequency set 2206. As anotherexample, a guided surface waveguide probe Pa . . . N may transmit atmultiple discrete frequencies 2212 within the critical load frequencyset 2206. As yet another example, multiple guided surface waveguideprobes Pa . . . N may transmit at different discrete frequencies 2212within the critical load frequency set 2206. Alternatively, to supplyelectrical energy to one or more loads that are classified asnon-critical, a guided surface waveguide probe Pa . . . N may transmitat one or more discrete frequencies 2212 within the non-critical loadfrequency set 2209 to transmit electrical energy that will be suppliedto a non-critical load.

In alternative embodiments, loads may be classified in varying degreesof criticality. For example, loads may be classified as high criticalityloads, medium criticality loads, low criticality loads, or non-criticalloads. Thus, the critical load frequency set 2206 may comprise severalfrequency sets 2103 a . . . N (FIG. 21), such as a high critical loadfrequency set, a medium critical load frequency set, and a low criticalload frequency set. Accordingly, each of the frequency sets 2103 a . . .N may be reserved for a predefined degree of criticality of one or moreloads. For example, a load that is classified as a high criticality loadmay comprise an emergency medical services critical load such as a loadassociated with an ambulance, a hospital, or a medical device. A loadthat is classified as a medium criticality load may be for example, anair conditioning load. A load that is classified as a low criticalityload may be for example, a lighting fixture load.

The guided surface waveguide probes Pa . . . N may transmit electricalenergy to one or more load at discrete frequencies 2212 within each ofthe frequency sets 2103 a . . . N. In some embodiments, the discretefrequencies 2212 within one or more frequency sets 2103 a . . . N arereserved for specific types of loads, such as loads within a specificsize range or loads within a specific degree of criticality. In variousembodiments, unused frequency sets 2203 a 2203 b, and 2203 c comprisefrequencies between frequency sets 2103 a . . . N, such as the criticalload frequency set 2206 and the non-critical load frequency set 2209. Insome embodiments, the unused frequency sets 2203 a, 2203 b, 2203 c, maybecome used if they are reserved. Each of the frequency sets 2103 a . .. N (FIG. 21) may comprise one or more reserved frequency sets 2215. Thereserved frequency sets 2215 may be dedicated to specific types of loadsand/or specific transmission classes 2106 a . . . N (FIG. 21).

As previously mentioned, the classification of loads may be determinedby whether the loads are critical loads or non-critical loads.Criticality may be determined by a type of load that is being suppliedwith electrical energy. For example, a medically critical load, such asa pacemaker or a life support machine, may be classified as a criticalload because if it is not supplied with electrical energy, a person'slife or health may be put at risk. Therefore, it may be more importantthat the medically critical load be supplied with electrical energycompared with the importance of supplying electrical energy to anothertype of load. As another example, a critical transportation load, suchas an emergency services vehicle such as ambulance or a fire truck, maybe classified as a critical load because if it is not supplied withelectrical energy, a person's life or health may be put at risk. As acontrary example, a load such as a smart watch, may be classified asnon-critical because it is not likely that a person's life or health isat risk if the smart watch does not receive electrical energy.

The classification of loads may additionally or alternatively bedetermined by a range of sizes of the loads. For example, loads may beclassified as small loads, medium loads, or large loads. As one example,a portable personal device may be classified as a small load andsupplied with electrical energy transmitted from a guided surfacewaveguide probe Pa . . . N operating in one or more of the transmissionclasses 2106 a . . . N (FIG. 21) corresponding to one or morefrequencies within one or more of the frequency sets 2103 a . . . N thatis necessary for the portable personal device to receive electricalenergy. Examples of portable personal devices include pacemakers,watches, mobile phones, mobile computing devices, etc. As anotherexample, an air conditioning unit may be classified as a medium load andsupplied with electrical energy transmitted from a guided surfacewaveguide probe Pa . . . N operating in one or more of the transmissionclasses 2106 a . . . N (FIG. 21) corresponding to one or morefrequencies within one or more of the frequency sets 2103 a . . . N thatis necessary for the air conditioning unit to receive electrical energy.Other examples of medium loads include household appliances such asrefrigerators, water pumps, etc. As yet another example, an automobilemay classified as a large load and supplied with electrical energytransmitted from a guided surface waveguide probe Pa . . . N operatingin one or more of the transmission classes 2106 a . . . N (FIG. 21)corresponding to one or more frequencies within one or more of thefrequency sets 2103 a . . . N that is necessary for the automobile toreceive electrical energy.

The classification of loads may additionally or alternatively bedetermined by whether a load being supplied with electrical energy ismobile or non-mobile. For example, a personal portable device, such as awatch or a cell phone may be classified as a mobile load. As anotherexample, a household appliance, such as a refrigerator, may beclassified as a non-mobile load. The classification of loads may also bedetermined by the nature of the load. For example, loads may beclassified as resistive, capacitive, inductive, or a combinationthereof. As one example, a light bulb may be classified as a resistiveload. As another example, an electric motor may classified as acapacitive load. As yet another example, a transformer may be classifiedas an inductive load. The nature of the load may also correspond to alevel or degree of criticality. The loads may also be classified by afunction of the load. For example, loads for lighting, loads forreceptacles, and loads for power may all be classified differently basedupon the function that each carries out. As one example, a light bulbmay classified as a lighting load. As another example, an item thatcouples to typical power outlets may be classified as a receptacle load.In various embodiments, the loads may be classified by which consumercategory they are associated. Consumer categories may includeresidential, commercial, industrial, and/or governmental or municipalcategories. As one example, a household appliance may be classified as aresidential load. As another example, an air conditioning unit for anoffice building may be classified as a commercial load. As anotherexample, a transformer may be classified as an industrial load. As yetanother example, loads for street lights may be classified as amunicipal load. Additionally, loads may vary in criticality depending onwhat consumer category that they are associated with. In someembodiments, each consumer category corresponds to a differenttransmission class. For example, loads for street lights, loads fordrainage purposes, and loads for water supply, which may all beclassified as governmental or municipal loads may be in a transmissionclass with a higher priority than a load to power a residential airconditioning unit.

Referring next to FIG. 23A, shown is an example of a transmissionspectrum 2109 which comprises one or more frequencies by which theguided surface waveguide probes Pa . . . N (FIG. 21) may be transmittingelectrical energy in the form of guided surface waves according tovarious embodiments. In this example, transmission spectrum 2109includes class A load high frequency set 2303 a, class B load mediumfrequency set 2303 b, and class C load low frequency set 2303 c.Additionally, transmission spectrum 2109 includes reserved frequency set2306 a, reserved frequency set 2306 b, and discrete frequencies 2212within the various frequency sets.

In the example, class A load high frequency set 2303 a includes thehighest frequency set within the transmission spectrum 2109. The class Aload high frequency set 2303 a may correspond to local frequency set2315 depicted in FIG. 23B. Class B load medium frequency set 2303 bincludes frequencies within the frequency set that are higher than thefrequencies in the class C load low frequency set and lower than thefrequencies in the class A load high frequency set. The class B loadmedium frequency set 2303 b may correspond to regional frequency set2312 depicted in FIG. 23B. The class C load low frequency set 2303 c mayinclude the lowest frequency set within the transmission spectrum 2109.The class C load low frequency set 2303 c may correspond to nationwide(or global) frequency set 2309 depicted in FIG. 23B.

According to various embodiments, guided surface wave receive structuresR may be assigned one or more discrete frequencies 2212 (FIG. 22) withinone or more frequency sets 2103 a . . . N (FIG. 21) based upon one ormore loads that are coupled to them. In this example, loads may beclassified according to a range of frequencies by which electricalenergy is transmitted. For example, Class A load high frequency set 2303a may be used to transmit electrical energy, for example, to loads forpersonal portable devices that receive electrical energy transmitted athigh frequencies. Thus, a guided surface wave receive structure R (FIG.20B) coupled to a load that receives electrical energy transmitted at ahigh frequency, may be assigned to one or more discrete frequencies 2212within the class A load high frequency set 2303 a. The class B loadmedium frequency set 2303 b may be used to transmit electrical energyto, for example, loads that service predefined geographic regions, suchas a utility plant. Thus, a guided surface wave receive structure Rcoupled to a load that receives electrical energy transmitted at mediumfrequency, may be assigned to one or more discrete frequencies 2212within the class B load medium frequency set 2303 b. The class C loadlow frequency set 2303 c may be used to transmit electrical energy totypes of loads that receive electrical energy via low frequencies. Thus,a guided surface wave receive structure R coupled to a load thatreceives electrical energy transmitted at low frequency, may be assignedto one or more discrete frequencies 2212 within the class C load lowfrequency set 2303 c.

Referring next to FIG. 23B, shown is a representation of the UnitedStates and a local frequency set 2315, regional frequency set 2312, andnationwide frequency set 2309. In this example, each of the transmissionclasses 2106 a . . . N (FIG. 21) may only transmit in a predefinedrange. Generally, the area of transmission is dictated by frequency andpotentially other factors. For example, a guided surface waveguide probePa . . . N (FIG. 21)transmitting electrical energy in a transmissionclass 2106 a . . . N corresponding to the local frequency set 2315 forexample, may only be configured to transmit in a predefined localgeographic area. As another example, a guided surface waveguide probe Pa. . . N (FIG. 21) transmitting electrical energy in a transmission class2106 a . . . N corresponding to the regional frequency set 2315 forexample, may only be configured to transmit in a predefined regionalgeographic area. As yet another example, a guided surface waveguideprobe Pa . . . N (FIG. 21) transmitting electrical energy in atransmission class 2106 a . . . N corresponding to the nationalfrequency set 2315 for example, may only be configured to transmit in apredefined national geographic area.

Referring next to FIG. 24, shown is a networked hierarchical powerdistribution system 2400 according to various embodiments. The networkedhierarchical power distribution system 2400 includes a transmissioncontrol system 2403 in data communication with one or more guidedsurface waveguide probes P and one or more receive controllers 2412 byway of network 2415. The network 2415 includes, for example, theInternet, intranets, extranets, wide area networks (WANs), local areanetworks (LANs), wired networks, wireless networks, or other suitablenetworks, etc., or any combination of two or more such networks.

The transmission control system 2403 may include, for example, one ormore server computers or one or more server banks, located in a singleinstallation or distributed among many different geographical locations.The transmission control system 2403 may also include one or moresensors and one or more data stores that store various data. The datastored in the data store, for example, is associated with the operationof the various functional entities described below.

The one or more of the guided surface waveguide probes P may operate inone or more of the transmission classes 2106 a . . . N (FIG. 21) at oneor more frequencies within one or more of the frequency sets 2103 a . .. N (FIG. 21) to transmit electrical energy in the form of guidedsurface waves to one or more guided surface wave receive structures R(FIG. 20B), which supply the electrical energy to one or more loads2409. Additionally or alternatively, multiple guided surface waveguideprobes P may transmit electrical energy to one guided surface wavereceive structure R. The one or more guided surface waveguide probes Pmay be coupled to one or more power sources 2418.

Each of the one or more guided surface wave receiver structures R maysupply electrical energy to one or more loads 2409. The one or moreguided surface wave receive structures R may be coupled to one or morereceive controllers 2412, which are in data communication with thetransmission control system 2403 by way of network 2415.

Next, a general description of the operation of the various componentsof the networked hierarchical power distribution system 2400 isprovided. To begin, the transmission control system 2403 may receive arequest for electrical energy from one or more receive controllers 2412coupled to one or more guided surface waveguide receive structures R(FIG. 20B) by way of network 2415. The transmission control system 2403may send a communication back to the one or more receive controllers2412 requesting identification of one or more loads to which theelectrical energy will be supplied. Alternatively, the one or morereceive controllers 2412 may identify one or more loads 2409 that theelectrical energy will be supplied to in the request. Additionally, theone or more receive controllers 2412 may state the class of the one ormore loads 2409 in the request. In some embodiments, the transmissioncontrol system 2403 accesses a record of each of the guided surface wavereceive structures R and each of the one or more loads 2409 in a datastore. For example, each of the guided surface wave receive structures Rand each of the loads 2409 may already be registered in a recordcontained in the data store. Thus, in some embodiments, the transmissioncontrol system 2406 may already have an identification of the one ormore loads 2409 that will be supplied with electrical energy.

Next, the transmission control system 2403 may assign each of the guidedsurface wave receive structures R to one or more discrete frequencies2212 (FIG. 22) within one or more frequency sets 2103 a . . . N (FIG.21). Additionally or alternatively, the transmission control system 2403may assign each of the guided surface wave receive structures R to oneor more discrete frequencies 2212 (FIG. 22) within one or more reservedfrequency sets 2215 (FIG. 21). As previously discussed, one or morefrequencies within the one or more frequency sets 2103 a . . . N (FIG.21) may correspond to one or more transmission classes 2106 a . . . N(FIG. 21), which are determined based at least in part on the one ormore loads 2409 that are being supplied with electrical energy. Thus,each of the guided surface wave receive structures R may be assigned toone or more discrete frequencies 2212 (FIG. 22) and/or one or morefrequency sets 2103 a . . . N (FIG. 21) based upon the one or more loads2409 to which each of the guided surface wave receive structures R iscoupled. The transmission control system 2403 may communicate theassignment to one or more receive controllers 2412 via a message overthe network 2415. During assignment of the guided surface wave receivestructures R, the transmission control system 2403 may evaluate whetherany of the power sources 2418 coupled to any of the guided surfacewaveguide probes P is overloaded or in danger of being overloaded. Insome embodiments, one or more of the guided surface waveguide probes Pmay already be transmitting electrical energy when a request is receivedby the transmission control system 2403. Thus, the transmission controlsystem 2403 may assign one or more guided surface wave receivestructures R to one or more guided surface waveguide probes P afterdetermining that none of the power sources 2418 coupled to the one ormore guided surface waveguide probes P is at risk of being overloaded orat overcapacity.

In other embodiments, the one or more guided surface wave receivestructures R may not be coupled to one or more receive controllers 2412,and may not be in data communication with any other components of thenetworked hierarchical power distribution system 2400 over the network2415. Thus, the transmission control system 2403 may assign one or morediscrete frequencies 2212 (FIG. 22) and/or one or more respectivefrequency sets 2103 a . . . N (FIG. 21) to the one or more guidedsurface wave receive structures R that are not in data communicationwith any other components of the networked hierarchical powerdistribution system 2400 over the network 2415. Alternatively oradditionally, one or more guided surface waveguide probes P may beassigned to transmit guided surface waves to the one or more guidedsurface wave receive structures R that are not in data communicationwith any other components of the networked hierarchical powerdistribution system 2400 over the network 2415. There is a possibilitythat the one or more guided surface waveguide probes P may becomeoverloaded. In response to a detection that one or more power sources2418 is at risk of being overloaded or in response to detecting that theone or more guided surface waveguide probes P does not have requisitegeneration capacity, one or more guided surface waveguide probes P maybe prevented from transmitting guided surface waves to the one or moreguided surface wave receive structures R that are not in datacommunication with any other components of the networked hierarchicalpower distribution system 2400 over the network 2415.

The transmission control system 2403 may also generate a record ofassignment of each of the guided surface wave receive structures R andeach load 2409 being supplied with electrical energy. Each of the guidedsurface wave receive structures R and each load 2409 may be identifiedwith a unique identification number that is stored in the data store.The record may also include the one or more discrete frequencies 2212(FIG. 22), the frequency sets 2103 a . . . N (FIG. 21), and thetransmission classes 2106 a . . . N (FIG. 21) that have been previouslyassigned and/or associated with each of the guided surface wave receivestructures R and each load 2409.

The transmission control system 2403 may also identify a priority ofpower distribution for each one of the guided surface wave receivestructures R and each load 2409. As previously discussed, loads 2409 maybe classified by a criticality or non-criticality of the load 2409.Accordingly, priority may be identified by a classification of the load2409 as a critical load or a non-critical load. For example, a loadclassified as a critical load may be assigned to one or more discretefrequencies 2212 (FIG. 22) with a higher priority for receivingelectrical energy over one or more discrete frequencies 2212 (FIG. 22)than a non-critical load assigned to a different one or more discretefrequencies 2212. In some embodiments, loads 2409 are classifiedaccording to a degree of criticality. There may be numerous degrees ofcriticality classifications for loads 2409. Accordingly, there may bemany levels of priority for receiving electrical energy among one ormore discrete frequencies 2212 (FIG. 22) and/or one or more frequencysets 2103 a . . . N (FIG. 21). In one embodiment, each degree ofcriticality corresponds to a level of priority. For example, a highestcriticality classification may correspond to a first priority. A mediumcriticality classification may correspond to a second priority. A lowcriticality classification may correspond to a third priority.

Additionally or alternatively, loads 2409 may be classified by a rangeof sizes. Thus, in various embodiments, priority for receivingelectrical energy may be determined by a range of sizes of the loads2409. In some embodiments the priority for receiving electrical energymay increase as the sizes of the loads 2409 increase. In otherembodiments, the priority for receiving electrical energy may increaseas the sizes of the loads 2409 decrease. Loads 2409 may also beclassified as mobile or non-mobile. In some embodiments, loads 2409 thatare mobile may have priority for receiving electrical energy over loads2409 that are non-mobile, while in other embodiments loads 2409 that arenon-mobile may have priority for receiving electrical energy over loads2409 that are mobile.

The transmission control system 2403 may reassign one or more loads 2409to another one or more discrete frequencies 2212 in response todetecting that one or more power sources 2418 is at risk of beingoverloaded or in response to detecting that one or more guided surfacewaveguide probes P does not have requisite generation capacity.Additionally or alternatively, the transmission control system 2403 maydetermine to cease transmitting electrical energy to one or more of theguided surface wave receive structures R and/or one or more of the loads2409 in response to detecting that one or more power sources 2418coupled to the one or more guided surface waveguide probes P is at riskof being overloaded. The transmission control system 2403 may selectwhich of the one or more of the guided surface wave receive structures Rand/or one or more of the loads 2409 will cease receiving electricalenergy based upon one or more classifications.

Referring next to FIG. 25, shown is a flowchart that provides oneexample of the operation of a portion of the transmission control system2403 (FIG. 24) according to various embodiments. It is understood thatthe flowchart of FIG. 25 provides merely an example of the manydifferent types of arrangements that may be employed to implement theoperation of the portion of the transmission control system 2403 asdescribed herein. As an alternative, the flowchart of FIG. 25 may beviewed as depicting an example of steps of a method implementedaccording to one or more embodiments.

Beginning with box 2503, the transmission control system 2403 associateseach of the guided surface waveguide probes P with at least onefrequency within a respective frequency set 2103 a . . . N (FIG. 21)corresponding to one of the transmission classes 2106 a . . . N (FIG.21). Each of the guided surface waveguide probes P may transmit withinmultiple frequencies within one of the respective frequency sets 2103 a. . . N (FIG. 21) or at a single frequency within one of the respectivefrequency sets 2103 a . . . N (FIG. 21).

Next, at box 2506, the transmission control system 2403 receives atleast one request for electrical energy over network 2415 (FIG. 24). Therequest may be received, for example, from one or more receivecontrollers 2412 (FIG. 24) coupled to one or more guided surface wavereceive structures R (FIG. 20B). The request may identify, for example,one or more loads 2409 (FIG. 24).

Next, at box 2509, electrical energy in the form of at least one guidedsurface wave is transmitted along a terrestrial medium to at least oneguided surface wave receive structure R. One or more loads 2409 may becoupled to the at least one guided surface wave receive structure R toreceive the electrical energy.

Moving on to FIG. 26, shown is a flowchart that provides on example ofassigning a frequency to a guided surface wave receive structure R (FIG.20B) according to various embodiments. Beginning at box 2603, thetransmission control system 2403 (FIG. 24) receives a request from areceive controller 2412 (FIG. 24) coupled to a guided surface wavereceive structure R. The request may identify a load 2409 (FIG. 24) thatis coupled to the guided surface wave receive structure R. Additionally,the request may identify a classification of the load 2409 and amagnitude of the load 2409. Alternatively, the request may identifymultiple loads 2409 that are coupled to the guided surface wave receivestructure R.

Next, at box 2606, the transmission control system 2403 determines allof the available frequencies for the guided surface wave receivestructure R to be assigned. In making the determination, factors toconsider may include for example, identifying a classification for theload 2409 coupled to the guided surface wave receive structure R, thesize of the load 2409, or other factors. By identifying a classificationof the load 2409, the transmission control system 2403 may determine afrequency set 2103 a . . . N (FIG. 21) and a transmission class 2106 a .. . N (FIG. 21) associated with the load 2409. Accordingly, thetransmission control system 2403 can determine one or more discretefrequencies 2212 (FIG. 22) within the frequency set 2103 a . . . N thatare available for use (i.e. that have capacity).

Next, at box 2609, the transmission control system 2403 may assign oneor more discrete frequencies 2212 to the guided surface wave receivestructure R. The transmission control system 2403 may instruct theguided surface wave receive structure R to tune into the one or morediscrete frequencies 2212 by communicating a message over a network 2415(FIG. 24). For example, the guided surface wave receive structures R maybe coupled to variable capacitors and/or inductors that may beelectronically changed by varying the tap positions on a coil or bymaking other adjustments as can be appreciated.

Finally, at box 2612, the transmission control system 2403 instructs oneor more guided surface waveguide probes P to transmit electrical energyin the form of at least one guided surface wave along a terrestrialmedium to the guided surface wave receive structure R.

Referring next to FIG. 27, shown is a schematic block diagram of thetransmission control system 2403 (FIG. 24) and the receive controller2412 (FIG. 24) according to an embodiment of the present disclosure. Thetransmission control system 2403 and the receive controller 2412 eachinclude one or more computing devices 2703. Each computing device 2703includes at least one processor circuit, for example, having a processor2706 and a memory 2709, both of which are coupled to a local interface2715. To this end, each computing device 2703 may comprise, for example,at least one server computer or like device. The local interface 2715may comprise, for example, a data bus with an accompanyingaddress/control bus or other bus structure as can be appreciated.

In the transmission control system 2403, stored in the memory 2709 areboth data and several components that are executable by the processor2706. In particular, stored in the memory 2709 and executable by theprocessor 2706 are the power distribution application 2715, andpotentially other applications. Also stored in the memory 2709 may be adata store 2712 and other data. In addition, an operating system may bestored in the memory 2709 and executable by the processor 2706.

In the receive controller 2412, stored in the memory 2709 are both dataand several components that are executable by the processor 2706. Inparticular, stored in the memory 2709 and executable by the processor2706 are the receive structure control application 2718, and potentiallyother applications. Also stored in the memory 2709 may be a data store2712 and other data. In addition, an operating system may be stored inthe memory 2709 and executable by the processor 2706.

It is understood that there may be other applications that are stored inthe memory 2709 and are executable by the processor 2706 as can beappreciated. Where any component discussed herein is implemented in theform of software, any one of a number of programming languages may beemployed such as, for example, C, C++, C#, Objective C, Java®,JavaScript®, Perl, PHP, Visual Basic®, Python®, Ruby, Flash®, or otherprogramming languages.

A number of software components are stored in the memory 2709 and areexecutable by the processor 2706. In this respect, the term “executable”means a program file that is in a form that can ultimately be run by theprocessor 2706. Examples of executable programs may be, for example, acompiled program that can be translated into machine code in a formatthat can be loaded into a random access portion of the memory 2709 andrun by the processor 2706, source code that may be expressed in properformat such as object code that is capable of being loaded into a randomaccess portion of the memory 2709 and executed by the processor 2706, orsource code that may be interpreted by another executable program togenerate instructions in a random access portion of the memory 2709 tobe executed by the processor 2706, etc. An executable program may bestored in any portion or component of the memory 2709 including, forexample, random access memory (RAM), read-only memory (ROM), hard drive,solid-state drive, USB flash drive, memory card, optical disc such ascompact disc (CD) or digital versatile disc (DVD), floppy disk, magnetictape, or other memory components.

The memory 2709 is defined herein as including both volatile andnonvolatile memory and data storage components. Volatile components arethose that do not retain data values upon loss of power. Nonvolatilecomponents are those that retain data upon a loss of power. Thus, thememory 2709 may comprise, for example, random access memory (RAM),read-only memory (ROM), hard disk drives, solid-state drives, USB flashdrives, memory cards accessed via a memory card reader, floppy disksaccessed via an associated floppy disk drive, optical discs accessed viaan optical disc drive, magnetic tapes accessed via an appropriate tapedrive, and/or other memory components, or a combination of any two ormore of these memory components. In addition, the RAM may comprise, forexample, static random access memory (SRAM), dynamic random accessmemory (DRAM), or magnetic random access memory (MRAM) and other suchdevices. The ROM may comprise, for example, a programmable read-onlymemory (PROM), an erasable programmable read-only memory (EPROM), anelectrically erasable programmable read-only memory (EEPROM), or otherlike memory device.

Also, the processor 2706 may represent multiple processors 2706 and/ormultiple processor cores and the memory 2709 may represent multiplememories 2709 that operate in parallel processing circuits,respectively. In such a case, the local interface 2715 may be anappropriate network that facilitates communication between any two ofthe multiple processors 2706, between any processor 2706 and any of thememories 2709, or between any two of the memories 2709, etc. The localinterface 2715 may comprise additional systems designed to coordinatethis communication, including, for example, performing load balancing.The processor 2706 may be of electrical or of some other availableconstruction.

Although power distribution application 2715, receive structure controlapplication 2718, and other various systems described herein may beembodied in software or code executed by general purpose hardware asdiscussed above, as an alternative the same may also be embodied indedicated hardware or a combination of software/general purpose hardwareand dedicated hardware. If embodied in dedicated hardware, each can beimplemented as a circuit or state machine that employs any one of or acombination of a number of technologies. These technologies may include,but are not limited to, discrete logic circuits having logic gates forimplementing various logic functions upon an application of one or moredata signals, application specific integrated circuits (ASICs) havingappropriate logic gates, field-programmable gate arrays (FPGAs), orother components, etc. Such technologies are generally well known bythose skilled in the art and, consequently, are not described in detailherein.

The flowcharts of FIGS. 25 and 26 show the functionality and operationof an implementation of portions of the transmission control system 2403and the receive controller 2412. If embodied in software, each block mayrepresent a module, segment, or portion of code that comprises programinstructions to implement the specified logical function(s). The programinstructions may be embodied in the form of source code that compriseshuman-readable statements written in a programming language or machinecode that comprises numerical instructions recognizable by a suitableexecution system such as a processor 2706 in a computer system or othersystem. The machine code may be converted from the source code, etc. Ifembodied in hardware, each block may represent a circuit or a number ofinterconnected circuits to implement the specified logical function(s).

Although the flowcharts of FIGS. 25 and 26 show a specific order ofexecution, it is understood that the order of execution may differ fromthat which is depicted. For example, the order of execution of two ormore blocks may be scrambled relative to the order shown. Also, two ormore blocks shown in succession in FIGS. 25 and 26 may be executedconcurrently or with partial concurrence. Further, in some embodiments,one or more of the blocks shown in FIGS. 25 and 26 may be skipped oromitted. In addition, any number of counters, state variables, warningsemaphores, or messages might be added to the logical flow describedherein, for purposes of enhanced utility, accounting, performancemeasurement, or providing troubleshooting aids, etc. It is understoodthat all such variations are within the scope of the present disclosure.

Also, any logic or application described herein, including the powerdistribution application 2715 and the receive structure controlapplication 2718, that comprises software or code can be embodied in anynon-transitory computer-readable medium for use by or in connection withan instruction execution system such as, for example, a processor 2706in a computer system or other system. In this sense, the logic maycomprise, for example, statements including instructions anddeclarations that can be fetched from the computer-readable medium andexecuted by the instruction execution system. In the context of thepresent disclosure, a “computer-readable medium” can be any medium thatcan contain, store, or maintain the logic or application describedherein for use by or in connection with the instruction executionsystem.

The computer-readable medium can comprise any one of many physical mediasuch as, for example, magnetic, optical, or semiconductor media. Morespecific examples of a suitable computer-readable medium would include,but are not limited to, magnetic tapes, magnetic floppy diskettes,magnetic hard drives, memory cards, solid-state drives, USB flashdrives, or optical discs. Also, the computer-readable medium may be arandom access memory (RAM) including, for example, static random accessmemory (SRAM) and dynamic random access memory (DRAM), or magneticrandom access memory (MRAM). In addition, the computer-readable mediummay be a read-only memory (ROM), a programmable read-only memory (PROM),an erasable programmable read-only memory (EPROM), an electricallyerasable programmable read-only memory (EEPROM), or other type of memorydevice.

Further, any logic or application described herein, including the powerdistribution application 2715 and the receive structure controlapplication 2718, may be implemented and structured in a variety ofways. For example, one or more applications described may be implementedas modules or components of a single application. Further, one or moreapplications described herein may be executed in shared or separatecomputing devices or a combination thereof. For example, a plurality ofthe applications described herein may execute in the same computingdevice 2703, or in multiple computing devices in the same computingenvironment. Additionally, it is understood that terms such as“application,” “service,” “system,” “engine,” “module,” and so on may beinterchangeable and are not intended to be limiting.

Disjunctive language such as the phrase “at least one of X, Y, or Z,”unless specifically stated otherwise, is otherwise understood with thecontext as used in general to present that an item, term, etc., may beeither X, Y, or Z, or any combination thereof (e.g., X, Y, and/or Z).Thus, such disjunctive language is not generally intended to, and shouldnot, imply that certain embodiments require at least one of X, at leastone of Y, or at least one of Z to each be present.

It should be emphasized that the above-described embodiments of thepresent disclosure are merely possible examples of implementations setforth for a clear understanding of the principles of the disclosure.Many variations and modifications may be made to the above-describedembodiment(s) without departing substantially from the spirit andprinciples of the disclosure. All such modifications and variations areintended to be included herein within the scope of this disclosure andprotected by the following claims.

It should be emphasized that the above-described embodiments of thepresent disclosure are merely possible examples of implementations setforth for a clear understanding of the principles of the disclosure.Many variations and modifications may be made to the above-describedembodiment(s) without departing substantially from the spirit andprinciples of the disclosure. All such modifications and variations areintended to be included herein within the scope of this disclosure andprotected by the following claims. In addition, all optional andpreferred features and modifications of the described embodiments anddependent claims are usable in all aspects of the disclosure taughtherein. Furthermore, the individual features of the dependent claims, aswell as all optional and preferred features and modifications of thedescribed embodiments are combinable and interchangeable with oneanother.

Therefore, the following is claimed:
 1. A method of distributing powerwithin a hierarchical power distribution system from a plurality ofguided surface waveguide probes to at least one guided surface wavereceive structure comprising: associating each of the plurality ofguided surface waveguide probes with a respective one of a plurality offrequency sets, each of the plurality of frequency sets corresponding toa transmission class among a plurality of transmission classes, whereineach of the plurality of guided surface waveguide probes is designatedto transmit within a respective transmission class among the pluralityof transmission classes; transmitting, by each of the plurality ofguided surface waveguide probes, at least one guided surface wave alonga terrestrial medium to the at least one guided surface wave receivestructure by generating at least one resultant field that synthesizes awave front incident at a complex Brewster angle of incidence of theterrestrial medium; and supplying, by the at least one guided surfacewave receive structure, electrical energy to at least one electricalload coupled to the at least one guided surface wave receive structure.2. The method of claim 1, wherein at least one characteristic isassociated with each of the plurality of transmission classes.
 3. Themethod of claim 2, wherein the at least one characteristic specifies arange of sizes of the at least one electrical load.
 4. The method ofclaim 3, wherein the at least one electrical load encompasses at leastone portable personal device.
 5. The method of claim 2, wherein the atleast one characteristic comprises a criticality of the at least oneelectrical load.
 6. The method of claim 5, wherein each of the pluralityof frequency sets comprises a plurality of discrete transmissionfrequencies reserved for a predefined criticality type of the at leastone electrical load.
 7. The method of claim 6, wherein the criticalitycomprises a medical criticality associated with at least one medicallycritical load, wherein the at least one medically critical load is vitalto a life of at least one entity associated with the at least onemedically critical load.
 8. The method of claim 7, wherein at least oneof the plurality of frequency sets includes a subset of frequenciesreserved for the medically critical loads.
 9. The method of claim 6,wherein the criticality comprises a transportation criticality.
 10. Themethod of claim 1, wherein each of the plurality of frequency setsservices electrical loads of a predefined type.
 11. The method of claim1, further comprising: receiving a request for power over a network fromthe at least one guided surface wave receive structure; and assigningthe at least one guided surface wave receive structure to a discretetransmission frequency within a respective one of the plurality offrequency sets.
 12. The method of claim 11, further comprising:generating a record including the assigning of the at least one guidedsurface wave receive structure to the discrete transmission frequencywithin the respective one of the plurality of frequency sets; anddetermining a priority of power distribution associated with the atleast one guided surface wave receive structure based at least in partupon the at least one electrical load.
 13. The method of claim 12,wherein the respective one of the plurality of frequency sets includes areserved frequency set, and the priority of power distribution isfurther based at least in part on whether the at least one guidedsurface wave receive structure is assigned to the reserved frequency setwithin the respective one of the plurality of frequency sets.
 14. Asystem comprising: at least one guided surface waveguide probeconfigured to generate at least one resultant field that synthesizes awave front incident at a complex Brewster angle of incidence of thelossy conducting medium, the at least one guided surface waveguide probebeing associated with at least one respective frequency set among aplurality of frequency sets, each of a plurality of transmission classesbeing associated with each of the plurality of frequency sets; whereinthe at least one guided surface waveguide probe is configured totransmit at least one guided surface wave along a terrestrial medium toat least one guided surface wave receive structure within a respectivetransmission class among the plurality of transmission classes; andwherein the at least one guided surface wave receive structure isconfigured to supply electrical energy to at least one electrical loadcoupled to the at least one guided surface wave receive structure. 15.The system of claim 14, wherein the at least one electrical load isexperienced as at least one load at an excitation source coupled to theat least one guided surface waveguide probe transmitting the at leastone guided surface wave.
 16. The system of claim 14, wherein one of thetransmission classes specifies a geographical transmission class. 17.The system of claim 16, wherein the geographical transmission class isat least one of a local class, a regional class, or a nationwide class.18. The system of claim 14, further comprising at least one controlsystem in communication with the at least one guided surface waveguideprobe, the control system configured to assign the at least one guidedsurface waveguide probe to the at least one guided surface wave receivestructure.
 19. The system of claim 14, further comprising at least onecontrol system in communication with the at least one guided surfacewaveguide probe, the control system configured to assign the at leastone guided surface wave receive structure to a respective frequency setassociated with the at least one guided surface waveguide probe.
 20. Thesystem of claim 19, wherein the respective frequency set is assignedbased at least in part upon power data associated with a power sourcethat is coupled to the at least one guided surface waveguide probe. 21.The system of claim 14, wherein at least one of the plurality offrequency sets comprises a reserved frequency set that is selected foruse in connection with a specific category of electrical loads.
 22. Asystem comprising: at least one guided surface wave receive structureconfigured to: receive electrical energy from at least one guidedsurface wave transmitted along a terrestrial medium from at least oneguided surface waveguide probe transmitting in a respective transmissionclass among a plurality of transmission classes, the at least one guidedsurface wave receive structure being assigned the transmission class;and supply the electrical energy to at least one electrical load coupledto the at least one guided surface wave receive structure; and at leastone control system configured to assign the at least one guided surfacewaveguide probe to the at least one guided surface wave receivestructure in response to a request for power, the at least one guidedsurface waveguide probe being configured to generate at least oneresultant field that synthesizes a wave front incident at a complexBrewster angle of incidence of the lossy conducting medium.
 23. Thesystem of claim 22, wherein the at least one control system is furtherconfigured to assign the at least one guided surface wave receivestructure to a respective frequency set associated with the at least oneguided surface waveguide probe.